Chapter 1.2, Problem 61E

Solution

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

Vertical asymptote: $x=2$ .

Horizontal asymptote: $y=0$ .

Given information: 

The function $g\left(x\right)=\frac{4x-4}{x^3-8}$ .

Calculation:

Graph of

  $g\left(x\right)=\frac{4x-4}{x^3-8}$

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

Putting the value of $x=0$ then $y=0.5$ .

And $x=1$ , $y=0$

And $x=2$ , $y=\infty$

Plot the graph:

  

By inspection of the graph, vertical asymptote is at $x=2$ and horizontal asymptote is at $y=0$ .

Algebraically, the vertical asymptote corresponds to the fact that $x=2$ is

excluded from the domain, as that would cause a division by zero.

To demonstrate the horizontal asymptote. Now, it's sufficient to say that, as $x$ approaches infinity, the denominator becomes increasingly larger than the numerator, and the quotient of the division approaches zero.

Therefore, horizontal asymptote: $y=0$ and vertical asymptote: $x=2$ .