How to solve limit problems by using the l'Hospital's rule: examples and their solutions.
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)
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.
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.
Then the limit of ex/2 goes to 1/2.