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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

    • 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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