Lesson I: Limits Part I
Prerequisites: Algebra I

Let's start by defining a limit: A limit is what a dependent variable (of a function) (f(x)) approaches as an independent variable (x) approaches n, where n is a constant so long as it is the same number from the positive and negative side (not entirely correct, but it will work for the purposes of this lesson).
Or, in other words: "the limit of f(x) as x approaches n is [answer]"
So you can say that the "limit of x² as x approaches 2 is 4"

Or, in proper notation (-> is an arrow)
lim_(x->2) x² = 4

Is this the same thing as plugging x in and solving?
No, well, not exactly
A limit is what x approaches, not what x is.

Take the piece-wise function f(x) = {x² if x !=0 | 1 if x=0}
lim_(x->0) {x² if x !=0 | 1 if x=0} = 0

Why?
Because even though f(x) is 1 when x = 0, as x gets closer and closer to 0, f(x) gets closer and closer to 0.

Pretty cool, right?

So how do I solve a limit?
Ready?
You simplify the equation as much as possible, cancelling out anything and everything, then you plug in n for x.

So what if you have a function where "f(x) = 1/x", so that the number as x approaches 0 is different coming from the positive side than coming from the negative side?
That's where one-sided limits come into play. One-sided limits are the same as normal limits, but they are specifically as x->n from either the positive side or the negative side. A normal limit is when the two one-sided limits are the same.
one-sided limits are denoted as follows:

lim_(x->0)⁺ f(x)
if being approached from the positive (right) side, and
lim_(x->0)^- f(x)
if being approached from the negative (left) side.

so

lim_(x->0)⁺ 1/x = infinity
and
lim_(x->0)^- 1/x = negative infinity

These are useful with holes and other things you find in graphical algebra.

So how do I solve a one-sided limit?

The best way, as I have found, is to look at the graph and see if has an asymptote. If it is, is it going towards infinity, or negative infinity? There's your answer. If there is a jump discontinuity, you have to get an x value as close as you can to n from the correct side, and estimate based on that.

In the next lesson: Limits II: Derivatives by Limits

Lesson II: Derivatives by Limits
Prerequisites: Algebra I and Lessons I

So we now understand what a limit is, right? Why are they so important? Because (at least for now) they help you find derivatives.

First, we must understand what a derivative is.
A derivative is simply a rate of change, or a change in one variable divided by a change in another. This is the same as slope, kind of.

Slope is defined as the change in f(x) in relation to change in x. This can be denoted as , where means change in [variable].

is also denoted as
is denoted as because it is arbitrary in many cases. X is arbitrary as well.

So, slope is the rate of change where X and are arbitrary.

A derivative is a slope at a point. what is , then? 0, right? But that can't happen because we can't devide by 0. That's where limits come into play.

Oh, I get it! It must be , right?

Exactly! In other words,

So how do I use this?

Take the function f(x)=x²

simplifies to

and then to

and then

and then we plug in "0" for

= 2x

We can then plug in the "x" coordinate of the point of which we want the derivative!

There you have it! That's the derivative of x²

Derivatives are denoted as follows:



where means the derivative of y with respect to x and f(x) is the function.

That is all I have to teach you in this lesson! Here are a few practice problems!

find:







I'm not that creative with practice problems. You should go do these: https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/defderdirectory/DefDer.html

Next up: The "Cheater" Way to do Derivatives!