Chapter 11.1, Problem 22E

To determine

To find: The value of the $\underset{x\to 0}{\lim }\frac{-x}{\tan x}$ using table feature of a graphing utility. Also confirm your result graphically.

Answer to Problem 22E

The value of the limits is $\underset{x\to 0}{\lim }\frac{-x}{\tan x}=-1$

Explanation of Solution

Given information:

Given expression of limit as $\underset{x\to 0}{\lim }\frac{-x}{\tan x}$

Let, $f\left(x\right)=\frac{-x}{\tan x}$

The value of the function at $x=-0.1$ is,

  $\begin{array}{c}f\left(-0.1\right)=\frac{-\left(-0.1\right)}{\tan \left(-0.1\right)}\\ =\frac{0.1}{-0.1003}\\ =-0.997\end{array}$

The value of the function at $x=-0.01$ is,

  $\begin{array}{c}f\left(-0.01\right)=\frac{-\left(-0.01\right)}{\tan \left(-0.01\right)}\\ =\frac{0.01}{-0.0100003}\\ =-0.99997\end{array}$

The value of the function at $x=-0.001$ is,

  $\begin{array}{c}f\left(-0.001\right)=\frac{-\left(-0.001\right)}{\tan \left(-0.001\right)}\\ =0.9999997\end{array}$

The value of the function at $x=0$ is,

  $\begin{array}{c}f\left(0\right)=\frac{-\left(0\right)}{\tan \left(0\right)}\\ =\frac{0}{0}\\ =\text{undefined}\end{array}$

The value of the function at $x=0.1$ is,

  $\begin{array}{c}f\left(0.1\right)=\frac{-\left(0.1\right)}{\tan \left(0.1\right)}\\ =-\frac{0.1}{0.100335}\\ =-0.99666\end{array}$

The value of the function at $x=0.01$ is,

  $\begin{array}{c}f\left(0.01\right)=\frac{-\left(0.01\right)}{\tan \left(0.01\right)}\\ =-\frac{0.01}{0.010000335}\\ =-0.999966\end{array}$

The value of the function at $x=0.001$ is,

  $\begin{array}{c}f\left(0.001\right)=\frac{-\left(0.001\right)}{\tan \left(0.001\right)}\\ =-\frac{0.001}{0.00100000033}\\ =-0.99999967\end{array}$

Hence, the complete table is,

  $\begin{array}{cccccccc}x& -0.1& -0.01& -0.001& 0& 0.1& 0.01& 0.001\\ f\left(x\right)& -0.997& -0.99997& -0.9999997& \text{undefined}& -0.99666& -0.999966& 0.99999967\end{array}$

The value of the limit is, $\underset{x\to 0}{\lim }\frac{-x}{\tan x}=-1$

Graph of the function can be drawn as,

  

From the graph it is clear that $\underset{x\to 0}{\lim }\frac{-x}{\tan x}=-1$