Chapter 1.2, Problem 60E

Solution

To find: all horizontal and vertical asymptotes of the function.

Horizontal asymptote: $y=0$ .

Given information: 

The function $P\left(x\right)=\frac{4}{x^2+1}$ .

Calculation:

Graph of

  $P\left(x\right)=\frac{4}{x^2+1}$

Let $y=p\left(x\right)$ .

Putting the value of $x=0$ then $y=4$ .

And $x=1$ , $y=2$

And $x=-1$ , $y=2$

And $x=-2$ , $y=0.8$

And $x=2$ , $y=0.8$

Plot the graph:

  

By inspection of the graph a horizontal asymptote is at $y=0$ .

To demonstrate the horizontal asymptote. Right now, it's sufficient to say that, $x$

approaches infinity, the denominator becomes increasingly larger than the numerator, and the quotient of the division approaches zero.

Therefore, horizontal asymptote: $y=0$ .