# 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
• 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?

• One does not solve limits; one evaluates limits. ${}\qquad{}$ – Michael Hardy Jul 31 '15 at 15:10
• Does the list seem too short? It depends what do you mean by "Algebra of limits". Is it only addition, multiplication and division? If so, you should add the Theorem of Limit of Composite Function and Theorem of Limit of Inverse function to your list. – Joseph Jul 31 '15 at 15:30
• This 13 February 2000 sci.math post of mine contains a summary of a hand-written handout on various limit methods that I used in the late 1990s when teaching very strong high school students at LSMSA. – Dave L. Renfro Jul 31 '15 at 15:57
• The list will get longer if it includes techniques to evaluate infinite series, products and continued fractions, and infinitely nested radicals, since they are respectively defined as the limits of finite series, products and continued fractions, and finitely nested radicals. – Vincenzo Oliva Jul 31 '15 at 16:17
• I think you have listed all the techniques used to evaluate limits. Perhaps you should mention explicitly the use of standard limits like $\lim\limits_{x \to 0}\dfrac{\sin x}{x} = 1$. I wonder what else could be used to evaluate a limit – Paramanand Singh Aug 1 '15 at 5:47

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$$