Chapter 13.2, Problem 26E

To determine

To find: the value of limit.

Answer to Problem 26E

The value of limit is $2$ .

Explanation of Solution

Given information:

The limit is $\underset{x\to 1}{\lim }\frac{x^8-1}{x^5-x}$ .

Calculation: to find the value of limit, first factorise and then divide out common factor and then simplify,

  $\begin{array}{l}\underset{x\to 1}{\lim }\frac{x^8-1}{x^5-x}=\underset{x\to 1}{\lim }\frac{\left(x^4+1\right)\left(x^2+1\right)\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)\left(x-1\right)\left(x^2+1\right)}\text{ [factorise]}\\ \underset{x\to 1}{\lim }\frac{x^8-1}{x^5-x}=\underset{x\to 1}{\lim }\frac{\left(x^4+1\right)}{x}\text{ [divide out common factor]}\\ \underset{x\to 1}{\lim }\frac{x^8-1}{x^5-x}=\frac{1+1}{1}\text{ [substitute]}\\ \underset{x\to 1}{\lim }\frac{x^8-1}{x^5-x}=2\text{ [simplify]}\end{array}$

By using graphing device, first put the limit in the device, then the value of limit is $2$ .