Chapter 11.2, Problem 58E

a.

To determine

To graphically approximate the limit of the function $\frac{x^4-1}{x^4-3x^2-4}$ as $x\to 2$ by using a graphing utility to graph the function.

Answer to Problem 58E

Limit does not exist.

Explanation of Solution

Given:

  $\underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}$

Graph:

  

As it is clear from the graph that at $x=2$ , the function is undefined therefore, limit does not exist.

b.

To determine

To numerically approximate the limitof the function $\frac{x^4-1}{x^4-3x^2-4}$ as $x\to 2$ using graphing utility.

Answer to Problem 58E

Limit does not exist.

Explanation of Solution

The numerically approximate value can’t be possible because limit does not exist.

c.

To determine

To evaluate the limit.

Answer to Problem 58E

  $\infty$

Explanation of Solution

Given:

  $\underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}$

Calculation:

  $\begin{array}{l}\\ \underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}\\ =\underset{x\to 2}{\lim }\frac{(x-1)(x^3+x^2+x+1)}{(x-2)(x^3+2x^2+x+2)}\\ =\underset{x\to 2}{\lim }\frac{(x-1)(x+1)(x^2-1)}{(x+2)(x-2)(x^2+1)}\\ =\frac{(2-1)(2+1)(4-1)}{(2+2)(2-2)(4+1)}\\ =\frac{9}{0}\\ =\infty \end{array}$