Limits of Indeterminate Forms and Improper Integrals

Hey guys, I have a few problems I need help with and answers checked.

**Problem 1.** Calculate each limit.

**a.**

**My Solution**

Form , Using L'Hopital's Rule.

**b.**

**My Solution**

Form , Using L'Hopital's Rule. = , Using L'Hopital's Rule.

c.

**My Solution**

I'm having trouble with this one.

Form , Using L'Hopital's Rule. ... This is also of the form . I've applied L'Hopital's Rule two more times and it just keeps getting nastier and nastier but is still keeping the form . Where am I going wrong?

**d.**

**My Solution**

Let

= Form , Using L'Hopital's Rule. = (Shortening the simplifying steps due to me not wanting carpal tunnel syndrome.) =

**Problem 2.** Determine whether the improper integral converges, or diverges. In the case of convergence, give its value.

a.

My Solution

u = lnx, du = dx, dv = dx, v = x

=

Here I don't know what I'm supposed to do. How am I supposed to get the limit if I have both x's and t's?

**b.**

**My Solution**

Here I run into the same problem as in 2a.

Any and all help would be much appreciated. Thanks in advance.