All these limits you posted here are of the same kind and all can be solved by using the Stolz-Cesaro theorem. I don't know why your teacher gave you these problems if he didn't show you this theorem.

Now, for the first one.

Let .

is a positive sequence, strictly ascending and unbounded.

.

Here i used the following result:

If then

Then .

The second one:

Let .

is a positive sequence, strictly ascending and unbounded.

.

So, .