Review the following concepts if needed:
 l’Hôpital’s Rule Help
 Other Indeterminate Forms Help
 Improper Integrals Help
 More on Improper Integrals Help
Indeterminate Forms Practice Test

If possible, use l’Hôpital’s Rule to evaluate each of the following limits. In each case, check carefully that the hypotheses of l’Hôpital’s Rule apply.

If possible, use l’Hôpital’s Rule to evaluate each of the following limits. In each case, check carefully that the hypotheses of l’Hôpital’s Rule apply.

If possible, use some algebraic manipulations, plus l’Hôpital’s Rule, to evaluate each of the following limits. In each case, check carefully that the hypotheses of l’Hôpital’s Rule apply.

Evaluate each of the following improper integrals. In each case, be sure to write the integral as an appropriate limit.

Evaluate each of the following improper integrals. In each case, be sure to write the integral as an appropriate limit.
Answers

(a) lim _{x → 0} (cos x − 1) = 0 and lim _{x → 0} x ^{2} − x ^{3} = 0 so l’Hôpital’s Rule applies. Thus
Now l’Hôpital’s Rule applies again to yield
(b) lim _{x →0} e ^{2 x} − 1 − 2 x = 0 and lim _{x → 0} x ^{2} + x ^{4} = 0 so l’Hôpital’s Rule applies. Thus
l’Hôpital’s Rule applies again to yield
(c) lim _{x →0} cos x ≠ 0, so l’Hôpital’s Rule does not apply. In fact the limit does not exist.
(d) lim _{x →1} [ln x ] ^{2} = 0 and lim _{x → 1} ( x − 1) = 0 so l’Hôpital’s Rule applies. Thus
(e) lim _{x →2} ( x − 2) ^{3} = 0 and lim _{x →2} sin( x − 2) − ( x − 2) = 0 so l’Hôpital’s Rule applies. Thus
Now l’Hôpital’s Rule applies again to yield
We apply l’Hôpital’s Rule one last time to obtain
(f) lim _{x →1} ( e ^{x} − 1) = 0 and lim _{x →1} ( x − 1) = 0 so l’Hôpital’s Rule applies. Thus

(a) lim _{x →1 + ∞} x ^{3} = lim _{x → + ∞} ( e ^{x} − x ^{2} ) = + ∞ so l’Hôpital’s Rule applies. Thus
l’Hôpital’s Rule applies again to yield
l’Hôpital’s Rule applies one more time to finally yield
(b) lim _{x → + ∞} ln x = lim _{x → + ∞} x = + ∞ so l’Hôpital’s Rule applies. Thus
(c) lim _{x → + ∞} e ^{− x} = lim _{x → + ∞} ln[ x / ( x + 1)] = 0 so l’Hôpital’s Rule applies. Thus
It is convenient to rewrite this expression as
Now l’Hôpital’s Rule applies once more to yield
We apply l’Hôpital’s Rule a last time to obtain
(d) Since lim _{x → + ∞} sin x does not exist, l’Hôpital’s Rule does not apply. In fact the requested limit does not exist.
(e) It is convenient to rewrite this limit as
Since lim _{x → − ∞} x = lim _{x → − ∞} e ^{− x} = ±∞, l’Hôpital’s Rule applies. Thus
(f) Since lim _{x → − ∞} ln  x  = lim _{x → −∞} e ^{− x} = + ∞, l’Hôpital’s Rule applies. Thus

(a) We write the limit as . Since lim _{x → + ∞} x ^{3} = lim _{x → + ∞} e ^{x} = + ∞, l’Hôpital’s Rule applies. Thus
We apply l’Hôpital’s Rule again to obtain
Applying l’Hôpital’s Rule one last time yields
(b) We write the limit as . Since lim _{x → + ∞} sin(1/ x ) = lim _{x → + ∞} 1/ x = 0, l’Hôpital’s Rule applies. Hence
(c) We rewrite the limit as . Since lim _{x → + ∞} ln[ x /( x + 1)] = lim _{x → + ∞} 1/( x + 1) = 0, l’Hôpital’s Rule applies. Thus
Now l’Hôpital’s Rule applies again and we obtain
(d) We rewrite the limit as . Since lim _{x → + ∞} ln x = lim _{x → + ∞} e ^{x} = + ∞, l’Hôpital’s Rule applies. Thus
(e) We write the limit as . Since lim _{x → − ∞} lim x ^{2} = lim _{x → −∞} e ^{−2 x} = 0, l’Hôpital’s Rule applies. Thus
l’Hôpital’s Rule applies one more time to yield
(f) We rewrite the limit as . Since lim _{x → 0} e ^{1/ x} = lim _{x →0} 1/ x = +∞, l’Hôpital’s Rule applies. Thus

4 . We do (a), (b), (c), (d).
Therefore
Now this equals
The second limit does not exist, so the original integral does not converage.

We do (a), (b), (c), (d).
Ask a Question
Have questions about this article or topic? AskRelated Questions
Q:
Q:
Q:
Q:
Popular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Definitions of Social Studies
 Grammar Lesson: Complete and Simple Predicates
 Child Development Theories
 Signs Your Child Might Have Asperger's Syndrome
 How to Practice Preschool Letter and Name Writing
 Social Cognitive Theory
 Theories of Learning