Limits are the basis for almost everything in calculus. By the time you get to derivatives, you'll be doing limits and you won't even know it. To become familiar with what a limit is, let's consider the function

$\frac{0}{0}$ is indeterminate, so the function $f(x)$ would be indeterminate at $x=1$. However, the function is defined at $x=0.9$ and $x=1.1$:

In fact, the function is defined for any $1\pm \delta,$ where $\delta \neq 0.$

In this lesson, we'll be considering the case where $\delta$ is very close, but not equal to $0.$

Try dragging the black point to $x=1$ in the above graph, and you will notice that the point disappears.

However, notice that it "looks like"

So, we can say that

The English for this is "As $x$ approaches 1, $f(x)$ approaches $2$."

Another possible interpretation is "The limit of $f(x)$ as $x$ approaches 1 is $2$."

This does not mean that $f(1)=2$ , but rather that if $x$ is very close to $1$, then is $f(x)$ very close to $2$.

The function covers almost all points covered by the function $x+1$.

We can confirm this by performing the following operation:

Important! This kind of math is only allowed in limits. Notice how we divided the top and bottom of the fraction by $x-1$, a non-constant – we only did this knowing that $x$ is very close to, but not equal to 1.

Now try this:

As we said before, it is okay to divide by non-constants in limits. So, we can say

We can just plug in $x=2$ here to find