# 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 x^{2} as x → 2^{+}.

Put 2^{+} into x^{2}.

2^{+} means

the number a little bit bigger than 2.

(almost the same)

So (2^{+})^{2} = 2^{2}.

2^{2} = 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.