Table of Limits

Last updated
Save as PDF

Page ID: 97324

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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)