Calculus II Assignment # 4

This assignment covers

Finding Limits using l'Hopital's Rule

Evaluating Indefinite Integrals

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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 now applying l'Hopital's rule to this fraction now we apply the rule one more time Since ln y = 0, y = e 0 = 1. The limit is 1. 6) reminder: e is a constant < 3. so we apply l'Hopital's rule we apply the rule again twice to get But 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 integrate we 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 integrand Now we split the integral at 0. If u = e x, du = e x dx, and e 2x +1 = u² + 1When we integrate, we get .

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