We are frequently interested in a function's “end behavior.” That is, what is the behavior of the function as the input variable increases without bound or decreases without bound.

Many times a function will approach a horizontal asymptote as its end behavior. If the horizontal asymptote $y=L$ represents the end behavior of the function $f$ both as $x$ increases without bound and as $x$ decreases without bound, we write $\lim\limits_{x\to\infty}\fe{f}{x}=L$ and $\lim\limits_{x\to-\infty}\fe{f}{x}=L\text{.}$

While working Limits and Continuity you investigated strategies for formally establishing limit values as $x\to\infty$ or $x\to-\infty\text{.}$ In this activity you are going to investigate a more informal strategy for determining these type limits.

Consider $\lim\limits_{x\to\infty}\frac{4x-2}{3+20x}\text{.}$ When the value of $x$ is really large, we say that the term $4x$ dominates the numerator of the expression and the term $20x$ dominates the denominator. We actually call those terms the dominant terms of the numerator and denominator. The dominant terms are significant because when the value of $x$ is really large, the other terms in the expression contribute almost nothing to the value of the expression. That is, for really large values of $x\text{:}$

\begin{align*} \frac{4x-2}{3+20x}&\approx\frac{4x}{20x}\\ &=\frac{1}{5} \end{align*}

For example, even if $x$ has the paltry value of $1000\text{,}$

\begin{align*} \frac{4(1000)-2}{3+20(1000)}&=\frac{3998}{20003}\\ &\approx0.19987\ldots\approx\frac{1}{5} \end{align*}

This tells us that $\lim\limits_{x\to\infty}\frac{4x-2}{3+20x}=\frac{1}{5}$ and that $y=\frac{1}{5}$ is a horizontal asymptote for the graph of $y=\frac{4x-2}{3+20x}\text{.}$

# Subsection9.6.1Exercises

The formulas used to graph Figures 9.6.19.6.5 are given below. Focusing first on the dominant terms of the expressions, match the formulas with the functions ($f_1$ through $f_5$).

##### 1

$y=\dfrac{3x+6}{x-2}$

##### 2

$y=\dfrac{16+4x}{6+x}$

##### 3

$y=\dfrac{6x^2-6x-36}{36-3x-3x^2}$

##### 4

$y=\dfrac{-2x+8}{x^2-100}$

##### 5

$y=\dfrac{15}{x-5}$

Use the concept of dominant terms to informally determine the value of each of the following limits.

##### 6

$\lim\limits_{x\to-\infty}\dfrac{4+x-7x^3}{14x^3+x^2+2}$

##### 7

$\lim\limits_{t\to-\infty}\dfrac{4t^2+1}{4t^3-1}$

##### 8

$\lim\limits_{\gamma\to\infty}\dfrac{8}{2\gamma^3}$

##### 9

$\lim\limits_{x\to\infty}\dfrac{(3x+1)(6x-2)}{(4+x)(1-2x)}$

##### 10

$\lim\limits_{t\to\infty}\dfrac{4e^t-8e^{-t}}{e^t+e^{-t}}$

##### 11

$\lim\limits_{t\to-\infty}\dfrac{4e^t-8e^{-t}}{e^t+e^{-t}}$