Chapter A3, Problem 11E

(a)

To determine

To find: the limit of the given function

Answer to Problem 11E

The limit of the given function is $L=0.841$

Explanation of Solution

Given:

  $f(x)=\sin x,c=1,\in =0.01$

Calculation:

The limit of the given function is

  $\underset{x\to 1}{\lim }\sin x=\sin 1\approx 0.841$

Conclusion:

Hence,the limit of the given function is $L=0.841$

(b)

To determine

To Find:The number $\delta >0$ such that for all $x$

Answer to Problem 11E

The number is $\delta =0.018$

Explanation of Solution

Given:

  $0<|x-c|<\delta \Rightarrow |f(x)-L|<\in$

Calculation:

Step: 1

  $\begin{array}{l}|\sin x-\sin 1|<0.01\\ \Rightarrow -0.01<\sin x-0.841<0.01\\ \Rightarrow 0.831<\sin x<0.851\\ \Rightarrow {\sin }^{-1}0.831<x<{\sin }^{-1}0.851\end{array}$

Step: 2

  $\begin{array}{l}|x-1|<\delta \\ \Rightarrow -\delta <x-1<\delta \\ \Rightarrow 1-\delta <x<1+\delta \\ \Rightarrow 1-\delta ={\sin }^{-1}0.831\Rightarrow \delta \approx 0.0182\\ Also1+\delta ={\sin }^{-1}0.851\Rightarrow \delta \approx 0.0188\end{array}$

Hence choose $\delta =0.018$

Conclusion:

Hence,the number is $\delta =0.018$