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

Contributors and Attributions

• Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

• Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)