Chapter 12.4, Problem 11E

To determine

To find: The limit of the given function.

Answer to Problem 11E

The limit of the given function is $-1$ .

Explanation of Solution

Given information:

The given function is,

  $\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\frac{1-x}{1+x}\right)$

Concept used:

The horizontal asymptote of the function is calculated as,

  $\begin{array}{l}\underset{x\to -\infty }{\lim }f\left(x\right)=L_1\\ \underset{x\to \infty }{\lim }f\left(x\right)=L_2\end{array}$

The given function is,

  $\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\frac{1-x}{1+x}\right)$

The limit of the given functionis calculated as,

  $\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\frac{1-x}{1+x}\right)\\ =\underset{x\to \infty }{\lim }\left(\frac{x\left(\frac{1}{x}-1\right)}{x\left(\frac{1}{x}+1\right)}\right)\\ =\frac{\left(\frac{1}{\infty }-1\right)}{\left(\frac{1}{\infty }+1\right)}\\ =-1\end{array}$

Consider the graph of the given function as,

  

Therefore, the limit of the given functionis $-1$ .