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