cal2-assign4

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
now
applying l'Hopital's rule to this fraction

now we apply the rule one more time
Since ln y = 0, y = e⁰ = 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 + 1
When we integrate, we get

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