Q. 18 - Calculus

Question

Use a calculator or other graphing utility to graph the function $f\left(x\right)=\frac{x-2}{x^2-x-2}$. 

(a) Show that f(x) is not defined at x = 2. How is this reflected in your calculator graph? 

(b) Use the graph to argue that even though f(2) is undefined, we have $\underset{x\to 2}{lim} f\left(x\right)=\frac{1}{3}$. 

Step-by-Step Solution

Answer

Part (a)

Part (b) $\frac{1}{3}$

Part (a) Step 1. Given information.

We have been given a function $f\left(x\right)=\frac{x-2}{x^2-x-2}$.

We have to use a calculator or other graphing utility to graph this function.

Show that f(x) is not defined at x = 2. How this is reflected in the calculator graph.

Part (a) Step 2. Sketch the graph.

Function is :

$f\left(x\right)=\frac{x-2}{x^2-x-2}$

The graph of the function is :

From the graph, we see that at x=2, there is no any particular value of the function,
Put x=2 in the function :

$\begin{array}{r}f\left(2\right)=\frac{2-2}{2^2-2-2}\\ =\frac{0}{4-4}\\ =\frac{0}{0}\end{array}$

                     = undefined

Part (b) Step 1. Solving the limit expression

Solving the limit expression, 

We get:

$\begin{array}{r}\underset{x\to 2}{lim} f\left(x\right)=\underset{x\to 2}{lim} \frac{x-2}{x^2-x-2}\\ =\underset{x\to 2}{lim} \frac{x-2}{x^2-2x+x-2}\\ =\underset{x\to 2}{lim} \frac{x-2}{x(x-2)+1(x-2)}\\ =\underset{x\to 2}{lim} \frac{(x-2)}{(x-2)(x+1)}\end{array}$

$\begin{array}{r}=\underset{x\to 2}{lim} \frac{1}{x+1}\\ =\frac{1}{2+1}\\ =\frac{1}{3}\end{array}$