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Limits are fundamental to calculus. They help us understand the behavior of functions as they approach specific points or infinity. Watch this introductory video to get started.

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Limit laws simplify the process of finding limits. Review the various laws, such as the Sum Law, Product Law, and Quotient Law, in this comprehensive guide.

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2.3 The Limit Laws - Calculus Volume 1 | OpenStax

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This free textbook is an OpenStax resource written to increase student access to high-quality, peer-reviewed learning materials.

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One-sided limits consider the behavior of a function as it approaches a point from one side—either the left or the right. See examples and detailed explanations here.

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Section 2.3 : One-Sided Limits

In the final two examples in the previous section we saw two limits that did not exist. However, the reason for each of the limits not existing was different for each of the examples.

We saw that

\[\mathop {\lim }\limits_{t \to 0} \,\,\cos \left( {\frac{\pi }{t}} \right)\]

did not exist because the function did not settle down to a single value as \(t\) approached \(t = 0\). The closer to \(t = 0\) we moved the more wildly the function oscillated and in order for a limit to exist the function must settle down to a single value.

However, we saw that

\[\mathop {\lim }\limits_{t \to 0} H\left( t \right)\hspace{0.25in}{\mbox{where,}}\hspace{0.25in}H\left( t \right) = \left\{ \begin{array}{ll}0 & {\mbox{if }}t < 0\\ 1 & {\mbox{if }}t \ge 0\end{array} \right.\]

did not exist not because the function didn't settle down to a single number as we moved in towards \(t = 0\), but instead because it settled into two different numbers depending on which side of \(t = 0\) we were on.

In this case the function was a very well-behaved function, unlike the first function. The only problem was that, as we approached \(t = 0\), the function was moving in towards different numbers on each side. We would like a way to differentiate between these two examples.

We do this with one-sided limits. As the name implies, with one-sided limits we will only be looking at one side of the point in question. Here are the definitions for the two one sided limits.

Right-handed limit

We say

\[\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = L\]

provided we can make \(f\left( x \right)\) as close to \(L\) as we want for all \(x\) sufficiently close to \(a\) with \(x > a\) without actually letting \(x\) be \(a\).

Left-handed limit

We say

\[\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = L\]

provided we can make \(f\left( x \right)\) as close to \(L\) as we want for all \(x\) sufficiently close to \(a\) with \(x < a\) without actually letting \(x\) be \(a\).

Note that the change in notation is very minor and in fact might be missed if you aren't paying attention. The only difference is the bit that is under the "lim" part of the limit. For the right-handed limit we now have \(x \to {a^ + }\)(note the "+") which means that we know will only look at \(x > a\). Likewise, for the left-handed limit we have \(x \to {a^ - }\)(note the "-") which means that we will only be looking at \(x < a\).

Also, note that as with the "normal" limit ( i.e. the limits from the previous section) we still need the function to settle down to a single number in order for the limit to exist. The only difference this time is that the function only needs to settle down to a single number on either the right side of \(x = a\) or the left side of \(x = a\) depending on the one‑sided limit we're dealing with.

So, when we are looking at limits it's now important to pay very close attention to see whether we are doing a normal limit or one of the one-sided limits. Let's now take a look at the some of the problems from the last section and look at one-sided limits instead of the normal limit.

Example 1 Estimate the value of the following limits. \[\mathop {\lim }\limits_{t \to {0^ + }} H\left( t \right)\hspace{0.25in}\,{\rm{and}}\hspace{0.25in}\mathop {\lim }\limits_{t \to {0^ - }} H\left( t \right)\hspace{0.5in}{\rm{where}}\hspace{0.25in}H\left( t \right) = \left\{ \begin{array}{ll}0 & {\mbox{if }}t < 0\\ 1 & {\mbox{if }}t \ge 0\end{array} \right.\]

Show Solution

To remind us what this function looks like here's the graph.

So, we can see that if we stay to the right of \(t = 0\) ( i.e. \(t > 0\)) then the function is moving in towards a value of 1 as we get closer and closer to \(t = 0\), but staying to the right. We can therefore say that the right-handed limit is,

\[\mathop {\lim }\limits_{t \to {0^ + }} H\left( t \right) = 1\]

Likewise, if we stay to the left of \(t = 0\) ( i.e \(t < 0\)) the function is moving in towards a value of 0 as we get closer and closer to \(t = 0\), but staying to the left. Therefore, the left-handed limit is,

\[\mathop {\lim }\limits_{t \to {0^ - }} H\left( t \right) = 0\]

In this example we do get one-sided limits even though the normal limit itself doesn't exist.

Example 2 Estimate the value of the following limits. \[\mathop {\lim }\limits_{t \to {0^ + }} \,\,\cos \left( {\frac{\pi }{t}} \right)\hspace{0.5in}\mathop {\lim }\limits_{t \to {0^ - }} \,\,\cos \left( {\frac{\pi }{t}} \right)\]

Show Solution

From the graph of this function shown below,

we can see that both of the one-sided limits suffer the same problem that the normal limit did in the previous section. The function does not settle down to a single number on either side of \(t = 0\). Therefore, neither the left-handed nor the right-handed limit will exist in this case.

So, one-sided limits don't have to exist just as normal limits aren't guaranteed to exist.

Let's take a look at another example from the previous section.

Example 3 Estimate the value of the following limits. \[\mathop {\lim }\limits_{x \to {2^ + }} g\left( x \right)\hspace{0.25in}{\rm{and}}\hspace{0.25in}\mathop {\lim }\limits_{x \to {2^ - }} g\left( x \right)\hspace{0.25in}\hspace{0.25in}{\rm{where}}\hspace{0.25in}g\left( x \right) = \left\{ \begin{array}{ll}\displaystyle \frac{{{x^2} + 4x - 12}}{{{x^2} - 2x}} & {\mbox{if }}x \ne 2\\ 6 & {\mbox{if }}x = 2\end{array} \right.\]

Show Solution

So, as we've done with the previous two examples, let's remind ourselves of the graph of this function.

In this case regardless of which side of \(x = 2\) we are on the function is always approaching a value of 4 and so we get,

\[\mathop {\lim }\limits_{x \to {2^ + }} g\left( x \right) = 4\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\mathop {\lim }\limits_{x \to {2^ - }} g\left( x \right) = 4\]

Note that one-sided limits do not care about what's happening at the point any more than normal limits do. They are still only concerned with what is going on around the point. The only real difference between one-sided limits and normal limits is the range of \(x\)'s that we look at when determining the value of the limit.

Now let's take a look at the first and last example in this section to get a very nice fact about the relationship between one-sided limits and normal limits. In the last example the one-sided limits as well as the normal limit existed and all three had a value of 4. In the first example the two one-sided limits both existed, but did not have the same value and the normal limit did not exist.

The relationship between one-sided limits and normal limits can be summarized by the following fact.

Fact

Given a function f(x) if,

\[\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = L\]

then the normal limit will exist and

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = L\]

Likewise, if

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = L\]

then,

\[\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = L\]

This fact can be turned around to also say that if the two one-sided limits have different values, i.e.,

\[\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) \ne \mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right)\]

then the normal limit will not exist.

This should make some sense. If the normal limit did exist then by the fact the two one-sided limits would have to exist and have the same value by the above fact. So, if the two one-sided limits have different values (or don't even exist) then the normal limit simply can't exist.

Let's take a look at one more example to make sure that we've got all the ideas about limits down that we've looked at in the last couple of sections.

Example 4 Given the following graph,

compute each of the following.

1. \(f( - 4)\)

2. \(\mathop {\lim }\limits_{x \to - {4^ - }} f\left( x \right)\)

3. \(\mathop {\lim }\limits_{x \to - {4^ + }} f\left( x \right)\)

4. \(\mathop {\lim }\limits_{x \to - 4} f\left( x \right)\)

1. \(f\left( 1 \right)\)

2. \(\mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right)\)

3. \(\mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right)\)

4. \(\mathop {\lim }\limits_{x \to 1} f\left( x \right)\)

1. \(f\left( 6 \right)\)

2. \(\mathop {\lim }\limits_{x \to {6^ - }} f\left( x \right)\)

3. \(\mathop {\lim }\limits_{x \to {6^ + }} f\left( x \right)\)

4. \(\mathop {\lim }\limits_{x \to 6} f\left( x \right)\)

Show Solution

a\(f( - 4)\) doesn't exist. There is no closed dot for this value of \(x\) and so the function doesn't exist at this point.

b\(\mathop {\lim }\limits_{x \to - {4^ - }} f\left( x \right) = 2\) The function is approaching a value of 2 as \(x\) moves in towards -4 from the left.

c\(\mathop {\lim }\limits_{x \to - {4^ + }} f\left( x \right) = 2\) The function is approaching a value of 2 as \(x\) moves in towards -4 from the right.

d\(\mathop {\lim }\limits_{x \to - 4} f\left( x \right) = 2\) We can do this one of two ways. Either we can use the fact here and notice that the two one-sided limits are the same and so the normal limit must exist and have the same value as the one-sided limits or just get the answer from the graph.

Also recall that a limit can exist at a point even if the function doesn't exist at that point.

e\(f\left( 1 \right) = 4\). The function will take on the \(y\) value where the closed dot is.

f\(\mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right) = 4\) The function is approaching a value of 4 as \(x\) moves in towards 1 from the left.

g\(\mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right) = - 2\) The function is approaching a value of -2 as \(x\) moves in towards 1 from the right. Remember that the limit does NOT care about what the function is actually doing at the point, it only cares about what the function is doing around the point. In this case, always staying to the right of \(x = 1\), the function is approaching a value of -2 and so the limit is -2. The limit is not 4, as that is value of the function at the point and again the limit doesn't care about that!

h\(\mathop {\lim }\limits_{x \to 1} f\left( x \right)\) doesn't exist. The two one-sided limits both exist, however they are different and so the normal limit doesn't exist.

i\(f\left( 6 \right) = 2\). The function will take on the \(y\) value where the closed dot is.

j\(\mathop {\lim }\limits_{x \to {6^ - }} f\left( x \right) = 5\) The function is approaching a value of 5 as \(x\) moves in towards 6 from the left.

k\(\mathop {\lim }\limits_{x \to {6^ + }} f\left( x \right) = 5\) The function is approaching a value of 5 as \(x\) moves in towards 6 from the right.

l\(\mathop {\lim }\limits_{x \to 6} f\left( x \right) = 5\) Again, we can use either the graph or the fact to get this. Also, once more remember that the limit doesn't care what is happening at the point and so it's possible for the limit to have a different value than the function at a point. When dealing with limits we've always got to remember that limits simply do not care about what the function is doing at the point in question. Limits are only concerned with what the function is doing around the point.

Hopefully over the last couple of sections you've gotten an idea on how limits work and what they can tell us about functions. Some of these ideas will be important in later sections so it's important that you have a good grasp on them.

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In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other.

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A function is continuous if it has no breaks, holes, or gaps. Read through this detailed explanation of the definition of continuity in calculus.

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Section 2.9 : Continuity

Over the last few sections we've been using the term "nice enough" to define those functions that we could evaluate limits by just evaluating the function at the point in question. It's now time to formally define what we mean by "nice enough".

Definition

A function \(f\left( x \right)\) is said to be continuous at \(x = a\) if

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]

A function is said to be continuous on the interval \(\left[ {a,b} \right]\) if it is continuous at each point in the interval.

Note that this definition is also implicitly assuming that both \(f\left( a \right)\) and \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) exist. If either of these do not exist the function will not be continuous at \(x = a\).

This definition can be turned around into the following fact.

Fact I

If \(f\left( x \right)\) is continuous at \(x = a\) then,

\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\hspace{0.5in}\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = f\left( a \right)\hspace{0.5in}\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = f\left( a \right)\]

This is exactly the same fact that we first put down back when we started looking at limits with the exception that we have replaced the phrase "nice enough" with continuous.

It's nice to finally know what we mean by "nice enough", however, the definition doesn't really tell us just what it means for a function to be continuous. Let's take a look at an example to help us understand just what it means for a function to be continuous.

Example 1 Given the graph of \(f\left( x \right)\), shown below, determine if \(f\left( x \right)\) is continuous at \(x = - 2\), \(x = 0\), and \(x = 3\).

Show Solution

First \(x = - 2\).

Now \(x = 0\).

Finally \(x = 3\).

From this example we can get a quick "working" definition of continuity. A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.

In other words, a function is continuous if its graph has no holes or breaks in it.

For many functions it's easy to determine where it won't be continuous. Functions won't be continuous where we have things like division by zero or logarithms of zero. Let's take a quick look at an example of determining where a function is not continuous.

Example 2 Determine where the function below is not continuous. \[h\left( t \right) = \frac{{4t + 10}}{{{t^2} - 2t - 15}}\]

Show Solution

A nice consequence of continuity is the following fact.

Fact 2

If \(f\left( x \right)\) is continuous at \(x = b\) and \(\mathop {\lim }\limits_{x \to a} g\left( x \right) = b\) then,

\[\mathop {\lim }\limits_{x \to a} f\left( {g\left( x \right)} \right) = f\left( {\mathop {\lim }\limits_{x \to a} g\left( x \right)} \right)\]

To see a proof of this fact see the Proof of Various Limit Properties section in the Extras chapter. With this fact we can now do limits like the following example.

Example 3 Evaluate the following limit. \[\mathop {\lim }\limits_{x \to 0} {{\bf{e}}^{\sin x}}\]

Show Solution

Another very nice consequence of continuity is the Intermediate Value Theorem.

Intermediate Value Theorem

Suppose that \(f\left( x \right)\) is continuous on \(\left[ {a,b} \right]\) and let \(M\) be any number between \(f\left( a \right)\) and \(f\left( b \right)\). Then there exists a number \(c\) such that,

All the Intermediate Value Theorem is really saying is that a continuous function will take on all values between \(f\left( a \right)\) and \(f\left( b \right)\). Below is a graph of a continuous function that illustrates the Intermediate Value Theorem.

As we can see from this image if we pick any value, \(M\), that is between the value of \(f\left( a \right)\) and the value of \(f\left( b \right)\) and draw a line straight out from this point the line will hit the graph in at least one point. In other words, somewhere between \(a\) and \(b\) the function will take on the value of \(M\). Also, as the figure shows the function may take on the value at more than one place.

It's also important to note that the Intermediate Value Theorem only says that the function will take on the value of \(M\) somewhere between \(a\) and \(b\). It doesn't say just what that value will be. It only says that it exists.

So, the Intermediate Value Theorem tells us that a function will take the value of \(M\) somewhere between \(a\) and \(b\) but it doesn't tell us where it will take the value nor does it tell us how many times it will take the value. These are important ideas to remember about the Intermediate Value Theorem.

A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations as the following example shows.

Example 4 Show that \(p\left( x \right) = 2{x^3} - 5{x^2} - 10x + 5\) has a root somewhere in the interval \(\left[ { - 1,2} \right]\).

Show Solution

\[p\left( x \right) = 0\]

So, we have,

\[p\left( c \right) = 0\]

Let's take a look at another example of the Intermediate Value Theorem.

Example 5 If possible, determine if \(f\left( x \right) = 20\sin \left( {x + 3} \right)\cos \left( {\frac{{{x^2}}}{2}} \right)\) takes the following values in the interval \([0,5]\).

1. Does \(f\left( x \right) = 10\)?

2. Does \(f\left( x \right) = - 10\)?

Show All Solutions Hide All Solutions
Show Discussion

a Does \(f\left( x \right) = 10\)? Show Solution

\[f\left( c \right) = 10\]

b Does \(f\left( x \right) = - 10\)? Show Solution

Okay, as the previous example has shown, the Intermediate Value Theorem will not always be able to tell us what we want to know. Sometimes we can use it to verify that a function will take some value in a given interval and in other cases we won't be able to use it.

For completeness sake here is the graph of \(f\left( x \right) = 20\sin \left( {x + 3} \right)\cos \left( {\frac{{{x^2}}}{2}} \right)\) in the interval [0,5].

From this graph we can see that not only does \(f\left( x \right) = - 10\) in [0,5] it does so a total of 4 times! Also note that as we verified in the first part of the previous example \(f\left( x \right) = 10\) in [0,5] and in fact it does so a total of 3 times.

So, remember that the Intermediate Value Theorem will only verify that a function will take on a given value. It will never exclude a value from being taken by the function. Also, if we can use the Intermediate Value Theorem to verify that a function will take on a value it never tells us how many times the function will take on the value, it only tells us that it does take the value.

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In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval.

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There are several types of discontinuities, including removable and non-removable. This video discusses their differences and how to identify them.

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The Intermediate Value Theorem is a key concept in calculus. It states that if a function is continuous on [a, b], then it takes on every value between f(a) and f(b). Learn more about it in this article.

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Intermediate value theorem (IVT) review (article) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Practice makes perfect! Here are some limit problems to test your understanding.

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Section 2.5 : Computing Limits

For problems 1 - 9 evaluate the limit, if it exists.

1. \(\mathop {\lim }\limits_{x \to 2} \left( {8 - 3x + 12{x^2}} \right)\) Solution

2. \(\displaystyle \mathop {\lim }\limits_{t \to \, - 3} \frac{{6 + 4t}}{{{t^2} + 1}}\) Solution

3. \(\displaystyle \mathop {\lim }\limits_{x \to \, - 5} \frac{{{x^2} - 25}}{{{x^2} + 2x - 15}}\) Solution

4. \(\displaystyle \mathop {\lim }\limits_{z \to 8} \frac{{2{z^2} - 17z + 8}}{{8 - z}}\) Solution

5. \(\displaystyle \mathop {\lim }\limits_{y \to 7} \frac{{{y^2} - 4y - 21}}{{3{y^2} - 17y - 28}}\) Solution

6. \(\displaystyle \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {6 + h} \right)}^2} - 36}}{h}\) Solution

7. \(\displaystyle \mathop {\lim }\limits_{z \to 4} \frac{{\sqrt z - 2}}{{z - 4}}\) Solution

8. \(\displaystyle \mathop {\lim }\limits_{x \to \, - 3} \frac{{\sqrt {2x + 22} - 4}}{{x + 3}}\) Solution

9. \(\displaystyle \mathop {\lim }\limits_{x \to 0} \frac{x}{{3 - \sqrt {x + 9} }}\) Solution

10. Given the function \[f\left( x \right) = \left\{ {\begin{array}{rc}{7 - 4x}&{x < 1}\\{{x^2} + 2}&{x \ge 1}\end{array}} \right.\]
Evaluate the following limits, if they exist.

1. \(\mathop {\lim }\limits_{x \to \, - 6} f\left( x \right)\)

2. \(\mathop {\lim }\limits_{x \to 1} f\left( x \right)\)

Solution

11. Given \[h\left( z \right) = \left\{ {\begin{array}{rc}{6z}&{z \le - 4}\\{1 - 9z}&{z > - 4}\end{array}} \right.\]
Evaluate the following limits, if they exist.

1. \(\mathop {\lim }\limits_{z \to 7} h\left( z \right)\)

2. \(\mathop {\lim }\limits_{z \to - 4} h\left( z \right)\)

Solution

For problems 12 & 13 evaluate the limit, if it exists.

1. \(\mathop {\lim }\limits_{x \to 5} \left( {10 + \left| {x - 5} \right|} \right)\) Solution

2. \(\displaystyle \mathop {\lim }\limits_{t \to \, - 1} \frac{{t + 1}}{{\left| {t + 1} \right|}}\) Solution

3. Given that \(7x \le f\left( x \right) \le 3{x^2} + 2\) for all x determine the value of \(\mathop {\lim }\limits_{x \to 2} f\left( x \right)\). Solution

4. Use the Squeeze Theorem to determine the value of \(\displaystyle \mathop {\lim }\limits_{x \to 0} {x^4}\sin \left( {\frac{\pi }{x}} \right)\). Solution

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Calculus I - Computing Limits (Practice Problems)

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Here is a set of practice problems to accompany the Computing Limits section of the Limits chapter of the notes for Paul Dawkins Calculus I course at Lamar University.

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Test your knowledge of continuity with these practice problems.

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Section 2-9 : Continuity

1. The graph of \(f\left( x \right)\) is given below. Based on this graph determine where the function is discontinuous.

Solution

2. The graph of \(f\left( x \right)\) is given below. Based on this graph determine where the function is discontinuous.

Solution

For problems 3 - 7 using only Properties 1 - 9 from the Limit Properties section, one-sided limit properties (if needed) and the definition of continuity determine if the given function is continuous or discontinuous at the indicated points.

For problems 8 - 12 determine where the given function is discontinuous.

1. \(\displaystyle f\left( x \right) = \frac{{{x^2} - 9}}{{3{x^2} + 2x - 8}}\) Solution

2. \(\displaystyle R\left( t \right) = \frac{{8t}}{{{t^2} - 9t - 1}}\) Solution

3. \(\displaystyle h\left( z \right) = \frac{1}{{2 - 4\cos \left( {3z} \right)}}\) Solution

4. \(\displaystyle y\left( x \right) = \frac{x}{{7 - {{\bf{e}}^{2x + 3}}}}\) Solution

5. \(g\left( x \right) = \tan \left( {2x} \right)\) Solution

For problems 13 - 15 use the Intermediate Value Theorem to show that the given equation has at least one solution in the indicated interval. Note that you are NOT asked to find the solution only show that at least one must exist in the indicated interval.

1. \(25 - 8{x^2} - {x^3} = 0\) on \(\left[ { - 2,4} \right]\) Solution

2. \({w^2} - 4\ln \left( {5w + 2} \right) = 0\) on \(\left[ {0,4} \right]\) Solution

3. \(4t + 10{{\bf{e}}^t} - {{\bf{e}}^{2t}} = 0\) on \(\left[ {1,3} \right]\) Solution

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Calculus I - Continuity (Practice Problems)

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Check your answers with these solutions to the limit and continuity practice problems.

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Here are a set of practice problems for the Calculus I notes. Click on the " Solution" link for each problem to go to the page containing the solution.

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section.

Here is a listing of sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.

Review - In this chapter we give a brief review of selected topics from Algebra and Trig that are vital to surviving a Calculus course. Included are Functions, Trig Functions, Solving Trig Equations and Equations, Exponential/Logarithm Functions and Solving Exponential/Logarithm Equations.

Functions - In this section we will cover function notation/evaluation, determining the domain and range of a function and function composition.
Inverse Functions - In this section we will define an inverse function and the notation used for inverse functions. We will also discuss the process for finding an inverse function.
Trig Functions - In this section we will give a quick review of trig functions. We will cover the basic notation, relationship between the trig functions, the right triangle definition of the trig functions. We will also cover evaluation of trig functions as well as the unit circle (one of the most important ideas from a trig class!) and how it can be used to evaluate trig functions.
Solving Trig Equations - In this section we will discuss how to solve trig equations. The answers to the equations in this section will all be one of the "standard" angles that most students have memorized after a trig class. However, the process used here can be used for any answer regardless of it being one of the standard angles or not.
Solving Trig Equations with Calculators, Part I - In this section we will discuss solving trig equations when the answer will (generally) require the use of a calculator ( i.e. they aren't one of the standard angles). Note however, the process used here is identical to that for when the answer is one of the standard angles. The only difference is that the answers in here can be a little messy due to the need of a calculator. Included is a brief discussion of inverse trig functions.
Solving Trig Equations with Calculators, Part II - In this section we will continue our discussion of solving trig equations when a calculator is needed to get the answer. The equations in this section tend to be a little trickier than the "normal" trig equation and are not always covered in a trig class.
Exponential Functions -In this section we will discuss exponential functions. We will cover the basic definition of an exponential function, the natural exponential function, i.e. \({\bf e}^{x}\), as well as the properties and graphs of exponential functions
Logarithm Functions - In this section we will discuss logarithm functions, evaluation of logarithms and their properties. We will discuss many of the basic manipulations of logarithms that commonly occur in Calculus (and higher) classes. Included is a discussion of the natural (\(\ln(x)\)) and common logarithm (\(\log(x)\)) as well as the change of base formula.
Exponential and Logarithm Equations - In this section we will discuss various methods for solving equations that involve exponential functions or logarithm functions.
Common Graphs - In this section we will do a very quick review of many of the most common functions and their graphs that typically show up in a Calculus class.

Limits - In this chapter we introduce the concept of limits. We will discuss the interpretation/meaning of a limit, how to evaluate limits, the definition and evaluation of one-sided limits, evaluation of infinite limits, evaluation of limits at infinity, continuity and the Intermediate Value Theorem. We will also give a brief introduction to a precise definition of the limit and how to use it to evaluate limits

Tangent Lines and Rates of Change - In this section we will introduce two problems that we will see time and again in this course : Rate of Change of a function and Tangent Lines to functions. Both of these problems will be used to introduce the concept of limits, although we won't formally give the definition or notation until the next section.
The Limit - In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections.
One-Sided Limits - In this section we will introduce the concept of one-sided limits. We will discuss the differences between one-sided limits and limits as well as how they are related to each other.
Limit Properties - In this section we will discuss the properties of limits that we'll need to use in computing limits (as opposed to estimating them as we've done to this point). We will also compute a couple of basic limits in this section.
Computing Limits - In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits.
Infinite Limits - In this section we will look at limits that have a value of infinity or negative infinity. We'll also take a brief look at vertical asymptotes.
Limits At Infinity, Part I - In this section we will start looking at limits at infinity, i.e. limits in which the variable gets very large in either the positive or negative sense. We will concentrate on polynomials and rational expressions in this section. We'll also take a brief look at horizontal asymptotes.
Limits At Infinity, Part II - In this section we will continue covering limits at infinity. We'll be looking at exponentials, logarithms and inverse tangents in this section.
Continuity - In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval.
The Definition of the Limit - In this section we will give a precise definition of several of the limits covered in this section. We will work several basic examples illustrating how to use this precise definition to compute a limit. We'll also give a precise definition of continuity..

Derivatives - In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, higher order derivatives and logarithmic differentiation.

The Definition of the Derivative - In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.
Interpretation of the Derivative - In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function.
Differentiation Formulas - In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers.
Product and Quotient Rule - In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate.
Derivatives of Trig Functions - In this section we will discuss differentiating trig functions. Derivatives of all six trig functions are given and we show the derivation of the derivative of \(\sin(x)\) and \(\tan(x)\).
Derivatives of Exponential and Logarithm Functions - In this section we derive the formulas for the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions - In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent.
Derivatives of Hyperbolic Functions - In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine.
Chain Rule - In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule!
Implicit Differentiation - In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.g. y = f(x) and yet we will still need to know what f'(x) is. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates (the next section).
Related Rates - In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work.
Higher Order Derivatives - In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.
Logarithmic Differentiation - In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients (sometimes easier than using product and quotient rule). More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i.e. there are variables in both the base and exponent of the function.

Applications of Derivatives - In this chapter we will cover many of the major applications of derivatives. Applications included are determining absolute and relative minimum and maximum function values (both with and without constraints), sketching the graph of a function without using a computational aid, determining the Linear Approximation of a function, L'Hospital's Rule (allowing us to compute some limits we could not prior to this), Newton's Method (allowing us to approximate solutions to equations) as well as a few basic Business applications.

Rates of Change - In this section we review the main application/interpretation of derivatives from the previous chapter (i.e. rates of change) that we will be using in many of the applications in this chapter.
Critical Points - In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions.
Minimum and Maximum Values - In this section we define absolute (or global) minimum and maximum values of a function and relative (or local) minimum and maximum values of a function. It is important to understand the difference between the two types of minimum/maximum (collectively called extrema) values for many of the applications in this chapter and so we use a variety of examples to help with this. We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter.
Finding Absolute Extrema - In this section we discuss how to find the absolute (or global) minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have.
The Shape of a Graph, Part I - In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative (or local) minimum and maximum values of a function and where a function will be increasing and decreasing. We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum.
The Shape of a Graph, Part II - In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points (but not all) as relative minimums or relative maximums.
The Mean Value Theorem - In this section we will give Rolle's Theorem and the Mean Value Theorem. With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter.
Optimization Problems - In this section we will be determining the absolute minimum and/or maximum of a function that depends on two variables given some constraint, or relationship, that the two variables must always satisfy. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc.
More Optimization Problems - In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section.
L'Hospital's Rule and Indeterminate Forms - In this section we will revisit indeterminate forms and limits and take a look at L'Hospital's Rule. L'Hospital's Rule will allow us to evaluate some limits we were not able to previously.
Linear Approximations - In this section we discuss using the derivative to compute a linear approximation to a function. We can use the lienar approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples.
Differentials - In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then.
Newton's Method - In this section we will discuss Newton's Method. Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. There are many equations that cannot be solved directly and with this method we can get approximations to the solutions to many of those equations.
Business Applications - In this section we will give a cursory discussion of some basic applications of derivatives to the business field. We will revisit finding the maximum and/or minimum function value and we will define the marginal cost function, the average cost, the revenue function, the marginal revenue function and the marginal profit function. Note that this section is only intended to introduce these concepts and not teach you everything about them.

Integrals - In this chapter we will give an introduction to definite and indefinite integrals. We will discuss the definition and properties of each type of integral as well as how to compute them including the Substitution Rule. We will give the Fundamental Theorem of Calculus showing the relationship between derivatives and integrals. We will also discuss the Area Problem, an important interpretation of the definite integral.

Indefinite Integrals - In this section we will start off the chapter with the definition and properties of indefinite integrals. We will not be computing many indefinite integrals in this section. This section is devoted to simply defining what an indefinite integral is and to give many of the properties of the indefinite integral. Actually computing indefinite integrals will start in the next section.
Computing Indefinite Integrals - In this section we will compute some indefinite integrals. The integrals in this section will tend to be those that do not require a lot of manipulation of the function we are integrating in order to actually compute the integral. As we will see starting in the next section many integrals do require some manipulation of the function before we can actually do the integral. We will also take a quick look at an application of indefinite integrals.
Substitution Rule for Indefinite Integrals - In this section we will start using one of the more common and useful integration techniques - The Substitution Rule. With the substitution rule we will be able integrate a wider variety of functions. The integrals in this section will all require some manipulation of the function prior to integrating unlike most of the integrals from the previous section where all we really needed were the basic integration formulas.
More Substitution Rule - In this section we will continue to look at the substitution rule. The problems in this section will tend to be a little more involved than those in the previous section.
Area Problem - In this section we start off with the motivation for definite integrals and give one of the interpretations of definite integrals. We will be approximating the amount of area that lies between a function and the \(x\)-axis. As we will see in the next section this problem will lead us to the definition of the definite integral and will be one of the main interpretations of the definite integral that we'll be looking at in this material.
Definition of the Definite Integral - In this section we will formally define the definite integral, give many of its properties and discuss a couple of interpretations of the definite integral. We will also look at the first part of the Fundamental Theorem of Calculus which shows the very close relationship between derivatives and integrals
Computing Definite Integrals - In this section we will take a look at the second part of the Fundamental Theorem of Calculus. This will show us how we compute definite integrals without using (the often very unpleasant) definition. The examples in this section can all be done with a basic knowledge of indefinite integrals and will not require the use of the substitution rule. Included in the examples in this section are computing definite integrals of piecewise and absolute value functions.
Substitution Rule for Definite Integrals - In this section we will revisit the substitution rule as it applies to definite integrals. The only real requirements to being able to do the examples in this section are being able to do the substitution rule for indefinite integrals and understanding how to compute definite integrals in general.

Applications of Integrals - In this chapter we will take a look at some applications of integrals. We will look at Average Function Value, Area Between Curves, Volume (both solids of revolution and other solids) and Work.

Average Function Value - In this section we will look at using definite integrals to determine the average value of a function on an interval. We will also give the Mean Value Theorem for Integrals.
Area Between Curves - In this section we'll take a look at one of the main applications of definite integrals in this chapter. We will determine the area of the region bounded by two curves.
Volumes of Solids of Revolution / Method of Rings - In this section, the first of two sections devoted to finding the volume of a solid of revolution, we will look at the method of rings/disks to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation.
Volumes of Solids of Revolution / Method of Cylinders - In this section, the second of two sections devoted to finding the volume of a solid of revolution, we will look at the method of cylinders/shells to find the volume of the object we get by rotating a region bounded by two curves (one of which may be the \(x\) or \(y\)-axis) around a vertical or horizontal axis of rotation.
More Volume Problems - In the previous two sections we looked at solids that could be found by treating them as a solid of revolution. Not all solids can be thought of as solids of revolution and, in fact, not all solids of revolution can be easily dealt with using the methods from the previous two sections. So, in this section we'll take a look at finding the volume of some solids that are either not solids of revolutions or are not easy to do as a solid of revolution.
Work - In this section we will look at is determining the amount of work required to move an object subject to a force over a given distance.

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Show Mobile Notice Show All Notes Hide All Notes Here are a set of practice problems for the Calculus I notes. Click on the " Solution" link for each problem to go to the page containing the solution.