Calculus Midoriさんのイラストまとめ

>Both ends of f(x)=x^2 approach positive infinity. They share the same end behavior.
>By the way, by the way! End behaviors are usually used in respect to positive and negative infinity.
>Moving on, it's different for f(x)=x^3.

>You should know that the end behavior of polynomials are determined by its leading coefficient's exponent.
>If it's even, the end behavior for both sides will be the same. If odd, they'll be different.
>Here, I have f(x)=x^2 and f(x)=x^3 as our wonderful little examples.

>They're like limits, except for the "ends" of the graph. In short, what they "act" like as they approach positive and negative infinity.
>...Huh? that's not good enough of an explanation? Fine, fine. I have time to spare anyway.

>That's the bare minimum of what there is to explain about limits. The technical side of learning how to take asymptotes, limits, and holes will be discussed at a later time.
>Let's skedaddle maddle on to end behaviors!

>For one-sided limits, we use this notation.
>The limit as x approaches 0- is negative infinity, and the limit as x approaches 0+ is positive infinity...
>And so onn.

>The left side of the graph as it approaches x=0 is different as the right side... how fascinating! See how the left side shoots down towards negative infinity while the right side shoots up to positive infinity?
>BANG! Sorry, I just wanted to make sure you're paying attention.

>Did you notice how that only was 3? Well, much later, we'll be breaking the function into piecewises in order to prove continuity.
>For non-infinity limits, there will be 2 different limits for an asymptote, the limit for the right and left side. They're called one-sided limits.

>Asymptotes help determine limits AND end behaviors, which I'll get to soon.
>There can be multiple limits in a function. In 1/x, there exists 4 limits.
>The limit as x approaches negative/positive infinity and as it approaches 0.

>Shin wants to become me. Shin wants to live, but never does. Simple as that!
>If we look at the right side of the 1/x graph, the function is desperately approaching the x-value of 0. There exists an asymptote at y=0, however, and it therefore can't ever be x=0.

>However, x-values like 0.1, 0.01, or 0.001 are perfectly fine! The x-value will infinitely approach to x=0, but will never be x=0. This also goes for their respective y-values.
>Informally, we can define asymptotes as "what the function WANTS to become."