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 [ex - 1] is
ex.

Derivative of ex

The derivative of the denominator 2x is
2.

Derivative of a Polynomial

So, by the l'Hospital's rule,
the limit of (ex - 1)/2x
is equal to
the limit of (ex)/2.

The limit of ex/2 as x → 0 is
1/2.

So 1/2 is the answer.