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.