Chapter 5.4, Problem 9E

To determine

Graph each function. Show any asymptotes as dashed lines.

Answer to Problem 9E

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

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

Explanation of Solution

Given :

The function is :

  $\mathrm{h}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2+1}{{\mathrm{x}}^2}$

Calculation:

Make a table of different values of function:

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

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 = $\mathrm{x}=0$