# l'Hospital's Rule

How to use the l'Hospital's Rule to find the limit value: formula, 2 examples, and their solutions.

## Formula

### Formula

The limit of f(x)/g(x)

is equal to

the limit of f'(x)/g'(x).

This is the l'Hospital's rule.

This law can be used

when the limit of f(x)/g(x) is in 0/0 form.

## Example 1

### Example

Previously, you've solved this example.

Let's use the l'Hospital's rule

to solve this example.

### Solution

The derivative of the numerator sin 4x is

(cos 4x)⋅4.

Derivative of sin x

Derivative of a Composite Function

The derivative of the denominator x is

1.

So, by the l'Hospital's rule,

the limit of (sin 4x)/x

is equal to

the limit of [(cos 4x)⋅4]/1.

As x → 0,

cos 4x → 1.

Cosine Values of Commonly Used Angles

So the limit of [(cos 4x)⋅4]/1 is

[1⋅4]/1.

[1⋅4]/1 = 4.

So 4 is the limit value.

## Example 2

### Example

You can use the limit of (e^x - 1)/x

to solve this example.

But now,

Let's use the l'Hospital's rule

to solve this example.

### Solution

The derivative of the numerator [e^{x} - 1] is

e^{x}.

Derivative of e^{x}

The derivative of the denominator 2x is

2.

Derivative of a Polynomial

So, by the l'Hospital's rule,

the limit of (e^{x} - 1)/2x

is equal to

the limit of (e^{x})/2.

The limit of e^{x}/2 as x → 0 is

1/2.

So 1/2 is the answer.