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# 1.3: Limits at Infinity and Horizontal Asymptotes

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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  $$\lim\limits_{x\rightarrow - \infty} -2x^3+3^2+1$$.

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

## 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 $$\lim\limits_{x\rightarrow -\infty} \dfrac{2x+3}{3x-2}$$.

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

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

$$=\dfrac{1}{2}.$$

Example $$\PageIndex{4}$$:

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

Solution:

Exercise $$\PageIndex{4}$$

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

$$\dfrac{-1}{2}$$.

## 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)