Most limits can (and should) be found by expanding the function into its Taylor series. For example, here sin(x) = x + O(x^3) using the big-O notation.
today during class my lecturer mentioned that some limits that L'Hospital's rule applies to and seem to be solvable only by using the rule, can be in fact used without differentiation. Could you give any hints on the example he gave us?
I personally see no advantage to evaluating limits using Taylor expansions as opposed to L'Hospital's Rule, as these require differentiation to be found also. You are expected to be able to perform some algebraic manipulation so that you can get the function into a form where the limits can be evaluated, as in Skeeter's post.
Thanks for clearing that up for me guys. I have another question: could you recommend any books to study Calculus in depth? I presume older textbooks are more precise and to the point. My calculus textbook barely mentions Taylor's series and does not have any challenging problems.