Chapter 2.6, Problem 26E

Solution

The asymptotes and intercepts of function and graph the function.

The vertical asymptotes, $x=0$ , $x=2$ and $x=-2$ , horizontal asymptotes, $y=0$ with intercept is $\left(0,\infty \right)$ .

Given:

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

Concept Used:

If a polynomial function in the form $y=\frac{f\left(x\right)}{g\left(x\right)}=\frac{\left(a_n{x}^n+\mathrm{....}\right)}{\left(b_m{x}^m+\mathrm{....}\right)}$ , then, vertical asymptotes is given by real zeros of denominator provided it should not be zero of numerator.

And,

The end behaviour asymptote given by $\underset{x\to {\infty }^{-}}{\lim }\frac{f\left(x\right)}{g\left(x\right)}\text{ or }\underset{x\to {\infty }^{+}}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is horizontal asymptotes.

The condition can be concluded as,

1) If $n<m$ , the end behaviour is horizontal asymptotes that is $y=0$ ,

2) If $n=m$ the end behaviour is horizontal asymptotes is horizontal asymptotes that is $y=\frac{a_n}{b_n}$

3) If $n>m$ ,the quotient $q\left(x\right)$ of $f\left(x\right)/g\left(x\right)$ is the end behaviour asymptotes, where $q\left(x\right)$ is given by $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$ with $r\left(x\right)$ is remainder of $f\left(x\right)/g\left(x\right)$ . Hence, no horizontal asymptotes.

And,

The x -intercept is given by zeros of numerator that are not zero of denominator. And y -intercept is given by $f\left(0\right)$ .

From using asymptotes and intercepts the graph can be drawn.

Calculation:

Consider the function, $h\left(x\right)=\frac{3}{x^3-4x}$ .

To find vertical asymptotes, find the zeros of denominator, so factorize the denominator $x^3-4x$ :

  $\begin{array}{c}{x}^3-4x=x\left(x^2-4\right)\\ =x\left(x-2\right)\left(x+2\right)\end{array}$

Thus, the zeros of denominator of $h\left(x\right)$ is $x=0$ , $x=2$ and $x=-2$ , Means the vertical asymptotes are $x=0$ , $x=2$ and $x=-2$ .

To find end behaviour asymptotes, find $m\text{ and }n$ compare, $h\left(x\right)=\frac{3}{x^3-4x}$ with $y=\frac{f\left(x\right)}{g\left(x\right)}=\frac{\left(a_n{x}^n+\mathrm{....}\right)}{\left(b_m{x}^m+\mathrm{....}\right)}$ .

Since, $n<m$ end behaviour asymptotes is horizontal asymptotes, $y=0$ .

Now find the intercept, the x -intercept is given by zeros of numerator that are not zero of denominator.

Since, the numerator is constant, no x -intercept is present.

And y -intercept is given by $h\left(0\right)$ , so from $h\left(x\right)=\frac{3}{x^3-4x}$ .

  $\begin{array}{c}h\left(0\right)=\frac{3}{0^3-4\left(0\right)}\\ =\frac{3}{0}\\ =\infty \end{array}$

Thus y -intercept is $y=\infty$ or $\left(0,\infty \right)$ .

Hence, using vertical asymptotes, $x=0$ , $x=2$ and $x=-2$ , horizontal asymptotes, $y=0$ with intercept is $\left(0,\infty \right)$

the graph obtain is as:

  

Conclusion:

The vertical asymptotes, $x=0$ , $x=2$ and $x=-2$ , horizontal asymptotes, $y=0$ with intercept is $\left(0,\infty \right)$ .