# data[].attributes.body
1. Calculus is a branch of math that you use to calculate the instantaneous rate of change. It differs from other branches of math because it does not just use one formula or equation to solve a problem, you have to solve for the formula, and then solve the problem.

# data[].attributes.body
In mathematics, a limit is the value a function 'approaches' as the input value becomes increasingly closer to a certain value

# data[].attributes.body
F(x) =  25
Limx->8

"Limx->8" Is the limit statement, and it means that you are approaching x = 8 from both sides. "f(x)" is the function of the graph you are using. "25" is the y-value of the limit.

# data[].attributes.body
Graphically: Determine the y value on the graph by estimating as function 'approaches' a certain x value
With tables: Plug in values that become increasingly closer to the desired value (from both sides) ie. lim x-->4 then plug in 3.9, 3.99, 3.999 and 4.1, 4.01, 4.001 etc. if the function seems to indicate they all relate to the same value, then the limit exists

# data[].attributes.headline
If f(2)=4, can you conclude anything about the limit of f(x) as x approaches 2?

# data[].attributes.subject
If f(2)=4, can you conclude anything about the limit of f(x) as x approaches 2?

# data[].attributes.body
You can not conclude if 4 is the limit or not because there is not enough information. If the line is continuous then 4 is the limit, but if it is not continuous then there could be a hole at 4.

# data[].attributes.headline
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)?

# data[].attributes.subject
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)?

# data[].attributes.body
If  f(x) as x approaches 2 = 4, then f(2)=4

# data[].attributes.body
When evaluating a limit graphically, it is important to consider the y value that corresponds to the function as it 'approaches' a certain x value

# data[].attributes.body
There are three parts of defining continuity:
1. The function is not defined at x = c
2. The Limit of F(x) does not exist at x = c
3. The limit of F(x) exists at x = c, but it is not equal to F(c)

# data[].attributes.body
Point/removable discontinuity is when the two-sided limit exists, but isn't equal to the function's value. These types of discontinuities present themselves as holes.
Example: ( x^2-16 / x^2+4 ) has a removable discontinuity (hole) at x=4

# data[].attributes.body
There are two types of non-removable discontinuities
1. Jump discontinuity is when the two-sided limit doesn't exist because the one-sided limits aren't equal.
2. Asymptotic/infinite discontinuity is when the two-sided limit doesn't exist because it's unbounded.

# data[].attributes.body
Example: any function that is fully simplified that has a denominator of x+4 would have a Non-Removable discontinuity at x= -4

# data[].attributes.body
The intermediate value theorem describes a key property of continuous functions: for any function that's continuous over the interval [a,b], the function will take any value between f(a) and

f(b) over the interval.

# data[].attributes.body
Click on link below

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.body
function with vertical asymptotes at x=6 and x=-2, and with a zero at x=3:

(x-3)
---------------
(x-6)(x+2)

# data[].attributes.headline
Rational Functions and Vertical Asymptotes

# data[].attributes.subject
Rational Functions and Vertical Asymptotes

# data[].attributes.body
Not all rational functions have vertical asymptotes, in order to have a vertical asymptote the denominator must be set equal to zero and solved. In some cases they can not be solved. For example f(x) = 1/(x^2 + 4) you set x^2 +4 = 0 then subtract 4 so you get x^2 = -4, now you have to take the square root of x^2 and -4, but you can not take the root of -4, meaning that this can not be solved and does not have a vertical asymptote.

# data[].attributes.body
F'(x) is derivative notation, and is read as "F prime of x." g'(8) = 10 means that g(8) is differentiable at 10

# data[].attributes.body
A function is differentiable at a point when there's a defined derivative at that point. Meaning  the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.

# data[].attributes.body
In order to prove if a function is differentiable, first you must determine if it is continuous. If it is discontinuous then it cannot be differentiable. If it is proven continuous then you find the limit from the left and right. If they are equal then the function is differentiable. Example:

# data[].attributes.body
https://docs.google.com/document/d/1YLU_19suRvOehz32HIjKsh34-gEGyZdvXAoGu_ZaQ6c/edit