Chapter 11.2, Problem 51E

To determine

Tocalculate : The limit, $\underset{x\to 1^{-}}{\lim }\frac{1}{x^3+1}$ by finding one-sided limits of the function, f(x). Also, draw the graph of the function.

Answer to Problem 51E

The limit $\underset{x\to 1^{-}}{\lim }\frac{1}{x^3+1}$ does not exists. The graph of the function is given by,

  

Explanation of Solution

Given information : The function is $f\left(x\right)=\frac{1}{x^3+1}$ .

Formula used : If the left hand limit and right hand limits are equal then $\underset{x\to x}{\lim }f\left(x\right)=L$ otherwise the limits does not exists.

Calculation : Draw the graph of the function f(x).

  

The graph has asymptote at $x=-1$ . So, the left hand limit and right hand limit are not equal.

Thus the limit $\underset{x\to 1^{-}}{\lim }\frac{1}{x^3+1}$ does not exists.