Chapter 1.2, Problem 57E

Solution

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

Vertical asymptote: $x=3$

Horizontal asymptote: $y=-1$ .

Given information: 

The function $g\left(x\right)=\frac{x+2}{3-x}$ .

Calculation:

Graph of

  $g\left(x\right)=\frac{x+2}{3-x}$

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

Putting the value of $x=1$ then $y=0.5$ .

And $x=2$ , $y=4$

And $x=-1$ , $y=0.25$

And $x=-2$ , $y=0$

And $x=3$ , $y=\infty$

Plot the graph:

  

By inspection of the graph, vertical asymptote is at $x=3$ and horizontal asymptote is at $y=-1$ .

Algebraically, the vertical asymptote corresponds to the fact that $x=3$ is excluded from the domain, as that would cause a division by zero.

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

approaches infinity, the $+2$ and $3$ in the function become irrelevant and the function

approaches $\frac{x}{-x}=-1$ .

Therefore, vertical asymptote: $x=3$ and horizontal asymptote: $y=-1$ .