or "from the right" (i.e. from values of x greater than c):

However, not all functions behave like this. Some functions approach different values
depending on whether we let x approach c from the left or from the right. For these
functions, the two-sided limit does not exist, and we can only find the one-sided
limit. Consider what happens to the following function as x approaches 3:

f (x) =

As x approaches 3 from the left, f (x) approaches 9. We call 9 the left-hand
limit of f (x) as x approaches 3, and we denote this as

f (x) = 9

As x approaches 3 from the right, f (x) approaches 11. We call 11 the right-
hand limit of f (x) as x approaches 3, and we denote this as

f (x) = 11

Because there is no single value that f (x) approaches when x approaches 3, we must
say that the standard two-sided limit, or
f (x)
does not exist. In general,
f (x)
exists only if
f (x) = f (x) = L.
In other words, the two-sided limit exists only if the left-hand and right-hand limits
both exist and are equal.

Solving for Limits Using Limit Rules

Now that you know what limits are, you should become familiar with certain rules that
allow you to manipulate and solve for them. Several of them should make intuitive
sense.

Rule 1: f (x) = f (c) if f (x) is a polynomial
function.
This means that if you are solving for the limit of a polynomial function at x = c, you
can just plug x = c into the function to find the limit. For example,