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Mathematics LibreTexts

Table of limits

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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)


Table of limits is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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