# L'Hospital's Rule

How to solve limit problems by using the l'Hospital's rule: examples and their solutions.

## Formula

The limit of *f*(*x*)/*g*(*x*) is equal to

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

You can use this rule when:*f*(*x*)/*g*(*x*) is in 0/0 or ∞/∞ form,

the limit of *f*'(*x*) is not oscillating

the limit of *g*'(*x*) is not oscillating,

and *g*'(*x*) ≠ 0.

And even some limit problems

that satisfy above conditions

cannot be solved by using this rule.

However, l'Hospital rule is still very useful

when solving calculus level limit problems.

(especially 0/0 form)

Indeterminate form (Part 2)

## Example 1

Previously, you've solved this example before.

Limits of trigonometric functions

Let's solve the same example

by using the l'Hospital's rule.

As *x* → 0,

sin 4*x* → 0 and *x* → 0.

So this limit is in 0/0 form.

Then differentiate

both of the numerator and the denominator.

(sin 4*x*)' = (cos 4*x*)⋅4

Derivative of sin *x*

Chain rule in differentiation

(*x*)' = 1

Power rule in differentiation (Part 1)

Then the limit of (cos 4*x*)⋅4/1 goes to 4.

As you can see,

you can get the same answer.

## Example 2

As *x* → 0,*e*^{x} - 1 → 0 and 2*x* → 0.

So this limit is in 0/0 form.

Then differentiate

both of the numerator and the denominator.

(*e*^{x} - 1)' = *e*^{x}

Derivative of *e*^{x}

(2*x*)' = 2

Power rule in differentiation (Part 1)

Then the limit of *e*^{x}/2 goes to 1/2.