The first concept we will look at is something called a *limit*. Limits quantify what happens to the values of a function as we approach a given point. This can be defined notationally as:

`$\lim_{x \rightarrow 6} f(x) = L$`

We can read this in simple terms as “the limit as *x* goes to 6 of f(x) approaches some value L”. To evaluate this limit, we take points increasingly closer and closer to *6*— as close to 6 as we can get, but not 6 itself! We then evaluate where the function is headed at those points.

If we look at the limit of a function as x approaches a value from one direction, this is called a *one-sided limit*. For example, we might look at the values of *f(5)*, *f(5.9)*, *f(5.999)* and see if they are trending towards the value of *f(6)*. This is represented as:

`$\lim_{x \rightarrow 6^{-}} f(x) = L$`

We read this as “the limit as x approaches 6 from the left side approaches some value L.”

Whereas if we looked at the values of *f(6.1)*, *f(6.01)*, *f(6.0005)* and see if they are trending towards *f(6)*, we would represent this as:

`$\lim_{x \rightarrow 6^{+}} f(x) = L$`

We read this as “the limit as *x* goes to 6 from the right side approaches some value L.”

This takes us to the final key concept about limits. The limit as *x* approaches 6 exists only if the limit as *x* approaches 6 from the left side is equal to the limit as *x* approaches 6 from the right side. This is written out as:

if

`$\lim_{x \rightarrow 6^{-}} f(x) = \lim_{x \rightarrow 6^{+}} f(x) = L$`

then

`$\lim_{x \rightarrow 6} f(x) = L$`

if

`$\lim_{x \rightarrow 6^{-}} f(x) \neq \lim_{x \rightarrow 6^{+}} f(x)$`

then

`$\lim_{x \rightarrow 6} f(x)\ does\ not\ exist$`

### Instructions

The animation to the right shows limit calculations for the function at *x=-4* and *x=2*. In this animation, take note of the following:

- At
*x=-4*, the limit as*x*approaches -4 does not exist. This is because the one-sided limits each approach a different value - A
*x=2*, the limit as*x*approaches 2 exists since both one-sided limits approach the same value.