Chapter 2.6, Problem 2E

Solution

To Calculate:

The domain of the function $f\left(x\right)=\frac{-3}{x-3}$ .

To Describe using Limits:

The behavior of $f$ at the value(s) of $x$ excluded from the domain of $f$ .

The domain of $f$ is $ℝ-\left\{3\right\}$ .

The function $f$ approaches $\infty$ as $x$ approaches $3$ from the left and it approaches $-\infty$ as $x$ approaches $3$ from the right.

Given:

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

Concepts Used:

A function which is a fraction of two polynomials is a known as a rational function.

A rational function has a domain of $ℝ$ except that the points at which the denominator is zero are excluded.

The left hand limit of a function $f\left(x\right)$ at $x=a$ is calculated as $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(a-h\right)$ where $h>0$ .

The right hand limit of a function $f\left(x\right)$ at $x=a$ is calculated as $\underset{x\to a^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(a+h\right)$ where $h>0$ .

Calculations:

Find the domain of $f$ .

The function $f\left(x\right)=\frac{-3}{x-3}$ is a fraction with the polynomial $-3$ as the numerator and the polynomial $x-3$ as the denominator. Thus, it is a rational function.

The denominator $x-3$ becomes zero when $x=3$ . Thus, the point $x=3$ must be excluded from its default domain $ℝ$ .

Thus, $ℝ-\left\{3\right\}$ is the domain of $f$

Describe the behavior of $f$ at the point $x=3$ .

Calculate the left hand limit of $f$ at the point $x=3$ :

  $\begin{array}{c}\underset{x\to 3^{-}}{\lim }\frac{-3}{x-3}=\underset{h\to 0}{\lim }\frac{-3}{3-h-3}\\ \underset{x\to 3^{-}}{\lim }\frac{-3}{x-3}=\underset{h\to 0}{\lim }\frac{-3}{-h}\\ \underset{x\to 3^{-}}{\lim }\frac{-3}{x-3}=\underset{h\to 0}{\lim }\frac{3}{h}\\ \underset{x\to 3^{-}}{\lim }\frac{-3}{x-3}\to \infty \end{array}$

Thus, the left hand limit of $f$ at the point $x=3$ approaches $\infty$ .

Calculate the right hand limit of $f$ at the point $x=3$ :

  $\begin{array}{c}\underset{x\to 3^{+}}{\lim }\frac{-3}{x-3}=\underset{h\to 0}{\lim }\frac{-3}{3+h-3}\\ \underset{x\to 3^{+}}{\lim }\frac{-3}{x-3}=\underset{h\to 0}{\lim }\frac{-3}{h}\\ \underset{x\to 3^{+}}{\lim }\frac{-3}{x-3}\to -\infty \end{array}$

Thus, the right hand limit of $f$ at the point $x=3$ approaches $-\infty$ .

Conclusion:

The domain of $f$ is $ℝ-\left\{3\right\}$ .

The function $f$ approaches $\infty$ as $x$ approaches $3$ from the left and it approaches $-\infty$ as $x$ approaches $3$ from the right.