IXL - L'Hospital's rule: indeterminate forms involving quotients (Calculus practice)

Evaluate the limit. Write an exact, simplified answer.

lim

→4

–

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question

Evaluate the limit. Write an exact, simplified answer.

lim

→∞

key idea

lim

→

(

)

(

)

has indeterminate form

lim

→

(

)

and

lim

→

(

)

lim

→

(

)

(

)

has indeterminate form

∞

lim

→

(

)

±∞

and

lim

→

(

)

±∞

L'Hospital's rule, or L'Hôpital's rule, states that if

lim

→

(

)

(

)

has indeterminate form

∞

the functions

(

)

and

(

)

are differentiable and

′

(

)

≠

on an open interval containing

except possibly at

and

lim

→

′

(

)

′

(

)

exists or is

±∞

then

lim

→

(

)

(

)

lim

→

′

(

)

′

(

)

Note: L'Hospital's rule is also valid for one-sided limits and limits approaching

±∞

solution

Find

lim

→∞

Start by finding the limits of the numerator and the denominator.

lim
x
→∞
3
x
=
∞
lim
x
→∞
ln
x
=
∞

Both limits approach ∞, so

lim

→∞

has indeterminate form

∞

Apply L'Hospital's rule. Differentiate the numerator and denominator and find the limit.

lim

→∞

lim

→∞

′

(

)

′

lim

→∞

–

lim

→∞

–

lim

→∞

∞

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Not feeling ready yet? These can help:

H.14Find derivatives using the chain rule I

H.9Find derivatives using the product rule I

B.5Find limits of polynomials and rational functions

D.2Determine end behavior of polynomial and rational functions

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