Introducion to limits

Limits allow us to talk about what happens as we approach a point on a function, while this may seem random the idea of a limit is one of the foundational pillars of calculus.

The idea is there are some pesky functions like $f(x) = (2x + x^2)/x$ which aren't defined somewhere, in this case $f(0)$ is not defined because it would cause division by zero.

Let's look at the graph

I've marked the problem point, we can't divide by zero! yet it's almost as if $f(x)$ is begging us to make it two at zero. We can divide by $x$ to get

Now obviously if we plug in $x = 0$ we'll get $2$ as desired, then we'd divide by zero! The solution is to say as $x$ approaches two, $f(x)$ approaches zero.

What does that mean? It means however close to $2$ I want you to get, you can find an $x$ close to zero that satisfies my request.

For example: If I wanted to get within $0.01$ of $2$ you might pick $x = 0.001$. No matter how close I want to get, you can get me that close