Chapter 12, Problem 3RE

To determine

Is it possible to reach the limit?

Answer to Problem 3RE

The limit of the function can be reached.

Explanation of Solution

Given information:

Complete the table and use the result to estimate the limit numerically. Determine whether it is possible to reach the limit.

  $\begin{array}{cccccccc}x& -.01& -0.01& -0.001& 0& 0.001& 0.01& 0.1\\ f\left(x\right)& & & & ?& & & \end{array}$

  $\underset{x\to 0}{\lim }\left[\left(1-e^{-x}\right)/x\right]$

Calculation:

The given function is,

  $f\left(x\right)=\frac{1-e^{-x}}{x}$

When, $x=-.01$

  $f\left(x\right)=\frac{1-e^{-x}}{x}$

  $\begin{array}{l}f\left(x\right)=\frac{1-e^{-\left(-0.1\right)}}{-0.1}\\ =1.0517\end{array}$

When,

  $\begin{array}{l}x=-0.01\\ f\left(x\right)=\frac{1-e^{-x}}{x}\\ f\left(x\right)=\frac{1-e^{-\left(-0.01\right)}}{-0.01}\\ =1.005\end{array}$

When,

  $\begin{array}{l}x=-0.001\\ f\left(x\right)=\frac{1-e^{-x}}{x}\\ f\left(x\right)=\frac{1-e^{-\left(-0.001\right)}}{-0.001}\\ =1.0005\end{array}$

When,

  $\begin{array}{l}x=0.001\\ f\left(x\right)=\frac{1-e^{-x}}{x}\\ f\left(x\right)=\frac{1-e^{-\left(0.001\right)}}{0.001}\\ =.9995\end{array}$

When,

  $\begin{array}{l}x=0.01\\ f\left(x\right)=\frac{1-e^{-x}}{x}\\ f\left(x\right)=\frac{1-e^{-\left(0.01\right)}}{0.01}\\ =.99502\end{array}$

  $\begin{array}{l}x=0.1\\ f\left(x\right)=\frac{1-e^{-x}}{x}\\ f\left(x\right)=\frac{1-e^{-\left(0.1\right)}}{0.1}\\ =.95163\end{array}$

Now substitute the calculated values in the table.

  $\begin{array}{cccccccc}x& -.01& -0.01& -0.001& 0& 0.001& 0.01& 0.1\\ f\left(x\right)& 1.0517& 1.005& 1.0005& 0& 0.9995& 0.99502& 0.95163\end{array}$

Now find the limit of the function when $x=0$ by taking the average value of the function adjacent sides of the question mark.

  $\frac{1.0005+0.9995}{2}=1$

The estimated limit of the function as $x$ approaches zero is $\boxed{\underset{x\to 0}{\lim }\frac{1-e^{-x}}{x}=1}$

Hence, the limit of the function can be reached.