
# Table of Limits


## Properties of Limits

• Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing a. Assume that L and M are real numbers such that $$\lim_{x→a}f(x)=L$$ and $$\lim_{x→a}g(x)=M$$. Let c be a constant. Then, each of the following statements holds:

 Sum law for limits: $$\displaystyle \lim_{x→a}(f(x)+g(x))=\lim_{x→a}f(x)+\lim_{x→a}g(x)=L+M$$ Difference law for limits: $$\displaystyle \lim_{x→a}(f(x)−g(x))=\lim_{x→a}f(x)−\lim_{x→a}g(x)=L−M$$ Constant multiple law for limits: $$\displaystyle \lim_{x→a}cf(x)=c⋅\lim_{x→a}f(x)=cL$$ Product law for limits: $$\displaystyle \lim_{x→a}(f(x)⋅g(x))=\lim_{x→a}f(x)⋅\lim_{x→a}g(x)=L⋅M$$ Quotient law for limits: $$\displaystyle \lim_{x→a}\frac{f(x)}{g(x)}=\frac{\lim_{x→a}f(x)}{\lim_{x→a}g(x)}=\dfrac{L}{M}$$ for M≠0 Power law for limits: $$\displaystyle \lim_{x→a}(f(x))^n=(\lim_{x→a}f(x))^n=L^n$$ for every positive integer n. Root law for limits: $$\displaystyle \lim_{x→a}\sqrt[n]{f(x)}=\lim_{x→a}\sqrt[n]{f(x)}=\sqrt[n]{L}$$ for all L if n is odd and for $$L≥0$$ if n is even.

## Basic Limits

 $$\displaystyle \lim_{x→a}x=a$$ $$\displaystyle \lim_{x→a}c=c$$, where $$c$$ is a constant $$\displaystyle \lim_{x \to 0^+}\frac{1}{x}=+∞$$ $$\displaystyle \lim_{x \to 0^-}\frac{1}{x}=-∞$$ $$\displaystyle \lim_{x \to 0}\frac{1}{x^2}=+∞$$ $$\displaystyle \lim_{x \to a} p(x)=p(a)$$, where $$p(x)$$ is a polynomial function. $$\displaystyle \lim_{x \to 0^+}\frac{|x|}{x}=+1$$ $$\displaystyle \lim_{x \to 0^-}\frac{|x|}{x}=-1$$ $$\displaystyle \lim_{x→±∞} k=k$$, where $$k$$ is a constant. $$\displaystyle \lim_{x→∞} x^n=\infty$$, for all $$n \in \mathbb{N}$$. $$\displaystyle \lim_{x→-∞} x^n=\infty$$, when $$n$$ is even. $$\displaystyle \lim_{x→∞} x^n= -\infty$$, when $$n$$ is odd. $$\lim_{x→±∞}a_nx^n+a_{n−1}x^n−1+…+a^1x+a^0=\lim_{x→±∞}a_nx^n.$$ $$\lim_{x→±∞} \displaystyle \frac{1}{x^n}=0$$, for all $$n \in \mathbb{N}$$.

## Trigonometry limits

 $$\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1$$ $$\displaystyle\lim_{x \to 0}\sin(1/x)$$ =DNE