Chapter 12.1, Problem 74E

To determine

To find $\underset{x\to 5}{\lim }f\left(x\right)$ . Whether $f\left(x\right)$ is defined at $x=5$ . The effect of domain of $f\left(x\right)$ at the existence of the limit as x approaches 5.

Answer to Problem 74E

The function is not defined at $x=5$ .

Explanation of Solution

Given information:

A function is given as

  $f\left(x\right)=\frac{x^3-3x-10}{x-5}$

Calculation:

Consider the following function:

  $f\left(x\right)=\frac{x^3-3x-10}{x-5}$

To graph the function using graphing utility TI-83, proceed as follows.

Press $Y$ and key in the function. The display will be

  

Press windows and choose the proper scale. The display will be

  

Now press graph. The display will be

  

Now press trace and move left-right key to move the cursor to the required point. The display will be

  

Hence, the limiting value of the function as $x$ approaches 4 is $\underset{x\to 4}{\lim }f\left(4\right)=6$

When $x$ approaches 5, the limit of the function is defined according to the graph where as by numerical substitution, it is not defined. The discrepancy can be attributed to the fact the there is common factor between numerator and denominator of the function.

  $\underset{x\to 4}{\lim }f\left(5\right)=7$

  

No

The function is not defined at $x=5$

No.

Because there is common factor between numerator and denominator of the function $\left(x,-5\right)$ .