Calculating limits the easy way Couple of month ago in school we started learning limits. There were all sorts of ways some pretty hard and over-kill for something simple. Then I stumbled across l'Hopital (guess I spelled it right) and then all the simple limits got so easy, doing them in matter of seconds. 
So my question is: 
Are there more theorems or rules (like l'Hopital or similar) that you can apply to limits to calculate them more easily?
 A: I believe two of the more useful rules for limit calculations apply to algebraic operations within an expression. In particular,
$$\lim_{x \rightarrow a} f(x) + g(x) = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x).$$
And
$$\lim_{x \rightarrow a} f(x)g(x) = \left(\lim_{x \rightarrow a} f(x)\right)\left(\lim_{x \rightarrow a} g(x)\right),$$
As long as limits on the right hand side of each equation exist.
A: As noted by Andre Nicolas in comments Taylor series are a very powerful technique to evaluate limits. Once you are familiar with Taylor series it is very easy (and fast too) to apply the technique to evaluate even very complicated limits. Moreover Taylor series also show how slow or fast (or in what manner) the limiting value is approached. However in order to use this approach you need to be familiar with manipulation of infinite series. Most importantly you need to be able to guess correctly about the number of terms required in a particular Taylor series for a specific situation.
The technique of L'Hospital's Rule is also powerful but it has two big drawbacks for beginners:


*

*Very rarely do beginners understand and check the exact hypotheses to apply this technique.

*It tends to mechanize the overall process of evaluating limits and students think that differentiating and plugging the value of $x$ is all that is there to evaluation of limits which sort of goes against the very nature of limits.


There is another drawback of L'Hospital's Rule (which I would say is minor) that sometimes differentiation leads to complicated expressions and perhaps multiple applications of this rule are needed (which may lead to even more complicated expressions).
It is much better for beginners to stick to the laws of algebra of limits and Squeeze theorem. These basic theorems are intuitive and easy to understand (even their proofs too) and help the student to think of evaluating limits as something very different from plugging the value of $x$. Moreover combined with a set of standard limits (limit formulas for various elementary functions) these rules can be used to evaluate even complicated limits which perhaps seem to require the use of L'Hospital's Rule or Taylor series.
