Chapter 12.4, Problem 6E

To determine

To match the graph of function $f(x)=\frac{x^2}{x^2+1}$ with given graphs

  

Answer to Problem 6E

The graph of $f(x)$ is graph ( A).

Explanation of Solution

Given information :

Given function $f(x)=\frac{x^2}{x^2+1}$ and graphs

  

As we have $f(x)=\frac{x^2}{x^2+1}$

  $\Rightarrow$ The denominator of $f(x)$ is doesn’t vanish for any value of $x$ . So vertical asymptotes does not exist.

Thus, the horizontal asymptotes of $f(x)$ is obtained by evaluation of limit of $f(x)=\frac{x^2}{x^2+1}$ as $x\to \pm \infty$

  $\begin{array}{l}\underset{x\to \pm \infty }{\lim }f(x)=\underset{x\to \pm \infty }{\lim }\frac{x^2}{x^2+1}\\ =\underset{x\to \pm \infty }{\lim }\frac{x^2}{x^2\left(1+\frac{1}{x^2}\right)}\\ =\underset{x\to \pm \infty }{\lim }\frac{1}{\left(1+\frac{1}{x^2}\right)}\\ \Rightarrow \underset{x\to \pm \infty }{\lim }f(x)=1\left(\because \underset{x\to \pm \infty }{\lim }\frac{1}{x^2}=0\right)\\ \end{array}$

Thus, the function $f(x)$ has horizontal asymptotes $y=1$ ,the only graph that has horizontal asymptotes $y=1$ is graph (A) .

Hence, the graph of $f(x)$ is graph (A).