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) 
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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. 
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 + 1 When we integrate, we get

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