L'Hospital's Rule

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).

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

Find the limit of the given expression. The limit of (sin 4x)/x as x goes to 0

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 4x → 0 and x → 0.

So this limit is in 0/0 form.

Then differentiate
both of the numerator and the denominator.

(sin 4x)' = (cos 4x)⋅4

Derivative of sin x

Chain rule in differentiation

(x)' = 1

Power rule in differentiation (Part 1)

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

As you can see,
you can get the same answer.

Example 2

Find the limit of the given expression. The limit of (e^x - 1)/2x as x goes to 0

As x → 0,
ex - 1 → 0 and 2x → 0.

So this limit is in 0/0 form.

Then differentiate
both of the numerator and the denominator.

(ex - 1)' = ex

Derivative of ex

(2x)' = 2

Power rule in differentiation (Part 1)

Then the limit of ex/2 goes to 1/2.