For the first you have to know the third one from this (but it is good to know all):
(the last provided that the denominator is non-zero). In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left, called an indeterminate form, may still exist — this depends on the functions and .
You can find proof for these (or if you can't I can send you one).
For the second one I suppose you meant . This is easy just use the definition:
If is a real function, then the limit of as approaches infinity is , denoted
if and only if for all there exists such that whenever .