List of techniques to evaluate limits? I'd like to make a complete list of techniques to evaluate a limit.


*

*Definition of the limit

*Continuous functions

*Algebra of limits

*Addition, multiplication, division

*Composition

*Inverse function

*Showing inequalities

*Squeeze theorem

*Rewriting, try to factor out common factors in numerator and
denominator

*Rationalizing the denominator

*Substitutions, in particular the $1/t$ substitution. 

*Use of derivatives, l'Hôpital's rule and Taylor series.

*If $\lim_{x\to a} f(x)=1$ and $\lim_{x\to a} g(x)=\infty$ then $$\lim_{x\to a} f(x)^{g(x)} = e^{\lim_{x\to a} g(x)[f(x)-1]}$$

*for $$0^0\quad and\quad \infty^0 \quad form \implies $$
$$\lim_{x\to a} f(x)^{g(x)}=e^{\lim_{x\to a}[g(x) \log_e{f(x)}]}$$
However the list seems so short. Are there any other good strategies or techniques to solve limits?

The list of techniques to evaluate a limit that you have given above covers most of the non-elementary limit problems. Below I have given a few results that will be helpful in evaluating certain limit problems$:$
$ \bullet \ $ Assume that $f:(-a,a) \setminus \{0\} \rightarrow \Re \ $. Then $\lim_{x\to0}f(x)=l \ $ if and only if $ \ \lim_{x\to0}f(\sin(x))=l. $
$\bullet \ $ Assume that $f:(-a,a) \setminus \{0\} \rightarrow \Re \ $ . If $ \lim_{x\to0}f(x)=l \ $ then $ \lim_{x\to0}f(\left \vert {x} \vert \right) = l. $
$\bullet \ $ $ lim_{x\to \infty} \left(1+\frac{1}{x}\right)^x = e$ 
$\bullet$ $ lim_{x\to 0} (1+x)^\frac{1}{x} = e $
