What is L'Hôpital's rule used for?

Answer 1
L'Hôpital's rule is a really useful tool for evaluating limits of an indeterminate form #0/0#, or, #oo/oo#.
The theorem states that if we have a limit of an indeterminate form #0/0#, or, #oo/oo#, then:
# lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f'(x))/(g'(x)) #

Providing the limit does actually exist.

Proof

A specific proof for the case #0/0# is fairly easy, if #f# and #g# are differentible at #a# and the derivatives are continuous at #a#.
In this case, #f(a)=g(a)=0#:
# lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f(x)-0)/(g(x)-0) #
# " " = lim_(x rarr a) (f(x)-f(a))/(g(x)-g(a)) #
# " " = lim_(x rarr a) ((f(x)-f(a))/(x-a)) / ((g(x)-g(a))/(x-a) #
# " " = (lim_(x rarr a) (f(x)-f(a))/(x-a)) / (lim_(x rarr a)(g(x)-g(a))/(x-a) #
# " " = (f'(a)) / (g'(a)) #
# " " = lim_(x rarr a) (f'(x)) / (g'(x)) # QED

Example

# L = lim_(x rarr 0) (e^x-1)/x #
If we put #x=0# we get an indeterminate form #0/0#, so we can apply L'Hôpital's rule to get:
# L = lim_(x rarr 0) (d/dx (e^x-1))/(d/dx x # # \ \ = lim_(x rarr 0) (e^x)/1 # # \ \ = lim_(x rarr 0) (e^x)/1 # # \ \ = 1/1 # # \ \ = 1 #
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Answer 2

L'Hôpital's rule is used to evaluate limits of indeterminate forms, where the numerator and denominator both approach zero or infinity. It allows us to find the limit by taking the derivative of the numerator and denominator and then evaluating the limit again.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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