One-Sided Limits

How to find the one-sided limits (left-hand limit and right-hand limit): definition, 2 examples, and their solutions.

Definition

Left-Hand Limit

The left-hand limit of f(x) means
the limit of f(x) as x goes to a
[from the left side].

So the left-hand limit
has a minus sign behind a:
a-.

a- means
the number a little bit smaller than a.
(almost the same)

Right-Hand Limit

The right-hand limit of f(x) means
the limit of f(x) as x goes to a
[from the right side].

So the right-hand limit
has a plus sign behind a:
a+.

a+ means
the number a little bit bigger than a.
(almost the same)

When Does a Limit Exist?

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

In other words,
if the left-hand limit and the right-hand limit
are not equal,
then the limit of f(x) does not exist.

Example 1

Example

Solution

First find the left-hand limit.

This is when x → 2-:
x < 2.

So the left-hand limit of f(x) is
the limit of (-x + 6) as x → 2-.

Put 2- into (-x + 6).

Limit of a Function

2- means
the number a little bit smaller than 2.
(almost the same)

So -(2-) = -2.

-2 + 6 = 4

So the left-hand limit is 4.

Next, find the right-hand limit.

This is when x → 2+:
x > 2.

So the right-hand limit of f(x) is
the limit of x2 as x → 2+.

Put 2+ into x2.

2+ means
the number a little bit bigger than 2.
(almost the same)

So (2+)2 = 22.

22 = 4

So the right-hand limit is 4.

The left-hand limit is 4.
The right-hand limit is 4.

So the left-hand limit and the right-hand limit
are equal: 4.

So the limit of f(x) exist: 4.

So 4 is the answer.

Example 2

Example

Solution

First find the left-hand limit.

This is when x → 0-.

So the left-hand limit of f(x) is
the limit of x(x + 1)/|x| as x → 0-.

This is when x → 0-:
x < 0.

So change |x| to -x.

Absolute Value

Cancel the x factors.

Then x(x + 1)/(-x) = -(x + 1).

Put 0- into -(x + 1).

0- means
the number a little bit smaller than 0.
(almost the same)

So 0- = 0.

-(0 + 1) = -1

So the left-hand limit is -1.

Next, find the right-hand limit.

This is when x → 0+.

So the right-hand limit of f(x) is
the limit of x(x + 1)/|x| as x → 0+.

This is when x → 0+:
x > 0.

So change |x| to x.

Absolute Value

Cancel the x factors.

Then x(x + 1)/x = x + 1.

Put 0+ into x + 1.

0+ means
the number a little bit bigger than 0.
(almost the same)

So 0+ = 0.

0 + 1 = 1

So the right-hand limit is 1.

The left-hand limit is -1.
The right-hand limit is 1.

So the left-hand limit and the right-hand limit
are not equal.

Then the limit of f(x) does not exist.

So the limit of f(x) does not exist.