Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.3: Limits at Infinity and Horizontal Asymptotes

  • Page ID
    10254
  • [ "stage:draft", "article:topic" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Think ouT Loud

    What do mean by \(x\) approaches \(\infty\)?

     We have learned about  \( \lim\limits_{x \to a}f(x) = L\),  where  \(a\) is a real number. In this section we  would like to explore  \(a\) yo be \(\infty\) or \( -\infty\)

    As a motivating example, consider \(f(x) = 1/x^2\), as shown in Figure 1.1. Note how, as \(x\) approaches \(\infty\),\(f(x)\) approaches closer to \(0\). That is,

    \(\lim\limits_{x\rightarrow \infty} \frac{1}{x^2}=0\)


    \(\text{FIGURE 1.1}\): Graphing \(f(x)=1/x^2\) \).

    Some Basic Results:

    Basic facts

    1. \(\lim\limits_{x\rightarrow \infty}  k=k \), where \( k\) is a constant.
    2. \(\lim\limits_{x\rightarrow -\infty}  k=k \), where \( k\) is a constant. 
    3. \(\lim\limits_{x\rightarrow \infty}  x^n=\infty. n \in \mathbb{N}.\)
    4. \(\lim\limits_{x\rightarrow -\infty}  x^n= \infty\), when \(n\) is even.
    5. \(\lim\limits_{x\rightarrow -\infty}  x^n= -\infty\), when \(n\) is odd.
    6. \(\lim\limits_{x\rightarrow  \infty}  a_nx^n+a_{n-1}x^{n-1}+.........+a_0= \lim\limits_{x\rightarrow \infty}  a_nx^n\).

    Example \(\PageIndex{1}\):

    Find  \(\lim\limits_{x\rightarrow \infty}  -2x^3+3^2+1\).

    Solution:

     \(\lim\limits_{x\rightarrow \infty}  -2x^3+3^2+1 =  \lim\limits_{x\rightarrow \infty}  -2x^3 =-(\infty)= -\infty \).

    Exercise \(\PageIndex{1}\)

    Find  \(\lim\limits_{x\rightarrow - \infty}  -2x^3+3^2+1\).

    Answer

     \(\lim\limits_{x\rightarrow -\infty}  -2x^3+3^2+1 =  \lim\limits_{x\rightarrow -\infty}  -2x^3 =-(-\infty)=\infty \).

    Rational Functions

    Basic ResultS

    1. \(\lim\limits_{x\rightarrow \infty}  \dfrac{1}{x^n}=0, n \in \mathbb{N}.\)
    2. \(\lim\limits_{x\rightarrow  -\infty}  \dfrac{1}{x^n}=0, n \in \mathbb{N}.\)

    Note

    Below we will show two ways of solving limits at infinity of rational functions.

    Example \(\PageIndex{2}\):

    Find \(\lim\limits_{x\rightarrow \infty}  \dfrac{2x+3}{3x-2}\).

    Solution:

    Method 1: Divide both numerator and denominator by highest power of \(x\)  of the polynomial in the denominator.

    \(\lim\limits_{x\rightarrow \infty}  \dfrac{2x+3}{3x-2}\)

    \(=\lim\limits_{x \rightarrow \infty}  \dfrac{(\dfrac{2x+3}{x})}{(\dfrac{3x-2}{x}) }\)

    \(=\lim\limits_{x \rightarrow \infty}  \dfrac{(\dfrac{2x}{x}+\dfrac{3}{x})}{(\dfrac{3x}{x}-\dfrac{2}{x})}\)

    \(=\lim\limits_{x\rightarrow \infty}  \dfrac{(2+\dfrac{3}{x})}{(3-\dfrac{2}{x})}\)

    \(=\dfrac{2}{3}.\)

    Method 2:

    \(\lim\limits_{x\rightarrow \infty}  \dfrac{2x+3}{3x-2}=\lim\limits_{x\rightarrow \infty}  \dfrac{2x}{3x} =\dfrac{2}{3}.\)

    Exercise \(\PageIndex{2}\)

    Find \(\lim\limits_{x\rightarrow -\infty}  \dfrac{2x+3}{3x-2}\).

    Answer

    \(\lim\limits_{x\rightarrow -\infty}  \dfrac{2x+3}{3x-2}=\lim\limits_{x\rightarrow -\infty}  \dfrac{2x}{3x} =dfrac{2}{3}.\)

    Note

    The following properties of absolute valued functions would be useful.

    \(|x|=\begin{cases} -x& if x<0 \\ x & if x\geq 0\end{cases}\) ​​​​​​.

     

    Example \(\PageIndex{3}\):

    Find \(\lim\limits_{x\rightarrow -\infty} \dfrac{x+9}{\sqrt{4x^2+3x+2}}\).

    Solution.

    \(\lim\limits_{x\rightarrow -\infty} \dfrac{x+9}{\sqrt{4x^2+3x+2}}\)

    \( =\lim\limits_{x\rightarrow -\infty} \dfrac{x}{\sqrt{4x^2}}\)

    \( =\lim\limits_{x\rightarrow -\infty} \dfrac{\dfrac{x}{|x|}}{\dfrac{\sqrt{4x^2}}  {|x|} }\)

    \( =\lim\limits_{x\rightarrow -\infty} \dfrac{\dfrac{x}{-x}}{\dfrac{\sqrt{4x^2}  }{\sqrt{ x^2}}}\), Note that \(  \sqrt{ x^2}=|x| \).

    \(=\dfrac{-1}{2}.\)

    Exercise \(\PageIndex{3}\)

    Find \(\lim\limits_{x\rightarrow \infty} \dfrac{x+9}{\sqrt{4x^2+3x+2}}\).

    Answer

    \(=\dfrac{1}{2}.\)

    Example \(\PageIndex{4}\):

    Find \( \lim\limits_{x\rightarrow \infty} \sqrt{x^2+3x}-\sqrt{x^2+4x} \).

    Solution:

    Exercise \(\PageIndex{4}\)

     Find \( \lim\limits_{x\rightarrow \infty} \sqrt{x^2+3x}-\sqrt{x^2+4x} \)

    Answer

    \(\dfrac{-1}{2}\).

    Horizontal Asymptotes

    Definition

    Let \(f\) be a real valued function and \(L\) be a real number. \(y=L\) is called a horizontal asymptote if either \(\lim\limits_{x\rightarrow -\infty}f(x)=L\) or \(\lim\limits_{x\rightarrow \infty} f(x)=L.\)

    Example \(\PageIndex{1}\):

    Find  the horizontal asymptote(s) if any of the function

    \(f(x)=\dfrac{x+9}{\sqrt{4x^2+3x+2}}\).

    Solution:

    \(y==\dfrac{-1}{2}\), and \(y==\dfrac{1}{2}\) are the horizontal asymptotes .

    See Example \(\PageIndex{3}\) and Exercise \(\PageIndex{3}\).

    Contributors

    • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/

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