Calc 1 HW. l’Hopital’s Rule

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University of Colorado-Boulder
MATH 1300 Homework 11
1. Find the limit. Use l’Ho^pital’s Rule if appropriate. If there is a more elementary method,
consider using it. If l’Ho^pital’s Rule does not apply, explain why.
lim (1 + )
2. Use l’Ho^pital’s Rule to help find the asymptotes of f . Then use them, together with the
information from f ‘ and f ” , to sketch the graph. Check your work with a graphing device or
f (x) = xe-x
3. Prove that
ln (x)
for any number p > 0. This shows that the logarithmic function approaches 8 more slowly
than any power of x.
4. If an object with mass m is dropped from rest, one model for its speed v after t seconds,
taking air resistance into account is
(1 – e-ct/m )
where g is the acceleration due to gravity and c is a positive constant. (In Calculus 2 we will
be able to deduce this equation from the assumption that the air resistance is proportional
to the speed of the object; c is the proportionality constant.)
(a) Calculate lim v. What is the meaning of this limit?
(b) For fixed t, use l’Ho^pital’s Rule to calculate lim+ v. What can you conclude about the
velocity of a falling object in a vacuum?
5. If f ‘ is continuous, f (2) = 0, and f ‘ (2) = 7, evaluate
f (2 + 3x) + f (2 + 5x)
University of Colorado-Boulder
MATH 1300 Homework 11
Optional Challenge Problem
This problem will help improve your algebra skills.
Graph the function f (x) = (x – 2)1/3 x2/3 . Include calculation of the first and second derivatives and
full analysis of the increasing/decreasing behavior, local extrema, concavity and inflection points,
and the end behavior.

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