Chapter 5.4, Problem 10E

To determine

Graph each function. Show any asymptotes as dashed lines.

Answer to Problem 10E

Horizontal asymptote = $\mathrm{y}=-1$

Vertical asymptote = $\mathrm{x}=0$

Explanation of Solution

Given :

The function is :

  $\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}$

Calculation:

Make a table of different values of function:

x	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}$
1	0
-1	0
$\mathrm{x}\to 0^{+}$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}\to +\mathrm{\infty }$
$\mathrm{x}\to 0^{-}$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}\to +\mathrm{\infty }$
$\mathrm{x}\to +\mathrm{\infty }$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}\to -1$
$\mathrm{x}\to -\mathrm{\infty }$	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1-{\mathrm{x}}^2}{{\mathrm{x}}^2}\to -1$

Plot the points and we get the graph of the function.

  

An asymptote is a value of a function that you can get very near to, but you can never reach.

So, the function has two asymptotes:

Horizontal asymptote = $\mathrm{y}=-1$

Vertical asymptote = $\mathrm{x}=0$