Chapter 2.5, Problem 57E

a.

To determine

To show:that the limit under certain assumptionsis $\underset{x\to \infty }{\lim }{e}^{\frac{-x}{10}}=0$ .

Answer to Problem 57E

  $\underset{x\to \infty }{\lim }{e}^{\frac{-x}{10}}=0$

Explanation of Solution

Given:

The given limit is $\underset{x\to \infty }{\lim }{e}^{\frac{-x}{10}}$ .

Calculation:

  $\begin{array}{l}\underset{x\to \infty }{\lim }{e}^{\frac{-x}{10}}=\underset{x\to \infty }{\lim }\frac{1}{e^{\frac{x}{10}}}\\ =\frac{1}{e^{\infty }}=0\end{array}$

Hence, the limit under certain assumptions is $\underset{x\to \infty }{\lim }{e}^{\frac{-x}{10}}=0$ .

b.

To determine

To graph:the given function on a common screen.

Answer to Problem 57E

  $x=23.03$ .

Explanation of Solution

Given:

The given function is $y=e^x\text{ and}y=0.1$ .

Calculation:

Now draw the graph of the $y=e^x\text{ and}y=0.1$ on a single screen.

Graph is shown below:

  

From the graph, it is found that $x=23.03$ on the single screen.

c.

To determine

To solve: the function in part (b) without graphing device.

Answer to Problem 57E

  $x=23.03$ .

Explanation of Solution

Given:

The given function is $y=e^{\frac{-x}{10}}$ .

Calculation:

  $\begin{array}{l}{e}^{\frac{-x}{10}}<0.1\\ e\frac{x}{10}>10\left\{\text{taking reciprocal on both the sides}\right\}\\ \ln e\frac{x}{10}>\ln 10\left\{\text{taking natural log on both the sides}\right\}\\ \Rightarrow \frac{x}{10}>\ln 10\\ \Rightarrow x<10\ln 10\left\{\text{multiply 10 on both the sides}\right\}\\ x<23.03\end{array}$

From the above calculation, it is found that $x=23.03$ .