One-Sided Limits

One-Sided Limits

How to find one-sided limits (left-hand limit and right-hand limit) and the limit of a function: examples and their solutions.

Example 1

The graph of y = f(x) is given below. 1. Find the limit of f(x) as x goes to 1-. 2. Find the limit of f(x) as x goes to 1+. 3. Find the limit of f(x) as x goes to 1.

x → 1- means
x goes to 1
from the left side.

This is why when (-) is added,
the limit is called the 'left-hand limit'.

As x goes to 1-,
(= x goes to 1 from the left side.)
f(x) goes to 2-.
(= f(x) goes to 2 from downward.)

You don't have to write the (-)
when writing the answer.
So 2 is the answer.

x → 1+ means
x goes to 1
from the right side.

This is why when (+) is added,
the limit is called the 'right-hand limit'.

As x goes to 1+,
(= x goes to 1 from the right side)
f(x) goes to 2.
(No (+) or (-).)

The left-hand limit of f(x), 2,
and the right-hand limit f(x), 2,
are equal.

Then the limit of f(x) is that equal value: 2.

The limit of f(x) as xa
has nothing to do with f(a).

See the given graph.
You can see that
the limit of f(x) as x → 1 is 2,
which has nothing to do with f(1): 4.

Example 2

The graph of y = f(x) is given below. 1. Find the limit of f(x) as x goes to 3-. 2. Find the limit of f(x) as x goes to 3+. 3. Find the limit of f(x) as x goes to 3.

As x goes to 3-
(= goes to 3 from the left side),
f(x) goes to 2.

As x goes to 3+
(= goes to 3 from the right side),
f(x) goes to 3+.
(= goes to 3 from upward.)

So 3 is the answer.

The left-hand limit of f(x), 2,
and the right-hand limit f(x), 3,
are not equal.

Then the limit of f(x) doesn't exist.

The limit of a function exist
if and only if
the left-hand limit and the right-hand limit are equal.