Calculus II Assignment # 4

This assignment covers

Finding Limits using l'Hopital's Rule

Evaluating Indefinite Integrals

.

Questions

Questions 1 - 7: Identify the indeterminate form, use l'Hopital's rule to find the limits.

Questions 8 - 12: Determine if the improper integral converges. If so, evaluate it.

 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

.

Solutions

 1) 2) 3) 4) 5) so nowapplying l'Hopital's rule to this fraction now we apply the rule one more timeSince ln y = 0, y = e 0 = 1. The limit is 1. 6) reminder: e is a constant < 3. so we apply l'Hopital's rulewe apply the rule again twice to getBut e - 3 < 0 so the fraction becomes 7) we combine the fractions Now we apply l'Hopital's rule Now apply l'Hopital's rule again 8)We integrate, evaluate and then apply limit. 9) When we integratewe get ln | x + 2 | with domain x > - 2. Since we can't integrate it over - º to 0, the integral diverges. 10) 11) now substitute u = ln x and du = dx/x which = - º 12) do some algebra on the integrandNow we split the integral at 0.If u = e x, du = e x dx, and e 2x +1 = u² + 1When we integrate, we get

.

.

(all content of the MathRoom Lessons © Tammy the Tutor; 2002 - 2005).