I am trying to grasp how to solve problems for convergence in probability and distribution. So far the main tools we are utilizing are: Slutsky's Theorem (Lemma), Delta Method and Multivariate Delta Method, C.L.T., W.L.L.N., Finding the limiting distribution, Asymptotic Distributions. My text doesn't seem to be giving me the insight I need, and I can understand my Prof in lectures, but I get to the Homework Problems and I am completely lost!

I know this is a long question but:

1). How do we approach these problems like if we have to prove convergence in probability or distribution?

2). How do we know what method to use, i.e. slutsky, delta, etc....

3). How do we set it up to finding the limiting distribution and how do we know when we have found it?

Sorry, I was feeling comfortable in my graduate mathematical statistics class before this topic came up and now i am feeling like incessantly!

Any help that you could give would be great for the clueless person here since I have a midterm coming up in two weeks !!!!