Chapter 12.2, Problem 32E

a.

To determine

To find: To approximate the limit by using a graphing utility to graph the function.

Answer to Problem 32E

The approximate limit by using a graphing utility to graph the given function does not exist.

Explanation of Solution

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

  

The above graph is the graphical representation of the given function with the limits $\underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}$ . From the graph we observe the limit does not exist.

Thus we can find the limit using graphing utility.

b.

To determine

To find: To numerically approximate the limit by using the table feature of the graphing utility to create a table.

Answer to Problem 32E

The numerically approximate limit using a table from the graph does not exist.

Explanation of Solution

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

From the graph we can create a table,

x	f(x)
1.9	-6.692
1.99	-74.188
1.999	-749.188
2	Error
2.001	750.813
2.01	75.813
2.1	8.317

Therefore we can conclude that numerically the limit of the given function does not exist.

c.

To determine

To find: To algebraically evaluate the limit by using the appropriate techniques.

Answer to Problem 32E

By evaluating the given limit algebraically the limit does not exist.

Explanation of Solution

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

Initially factor the numerator and denominator,

  $\begin{array}{l}\underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}=\underset{x\to 2}{\lim }\frac{\left(x^2-1\right)\left(x^2-1\right)}{\left(x^2-4\right)\left(x^2+1\right)}\\ \underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}=\underset{x\to 2}{\lim }\frac{x^2-1}{x^2-4},\text{ by using direct substitution}\\ \underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}=\frac{2^2-1}{2^2-4}\\ \underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}=\frac{3}{0}\\ \underset{x\to 2}{\lim }\frac{x^4-1}{x^4-3x^2-4}=\text{undefined}\end{array}$

Thus we can approximate the limit algebraically.