Chapter 9.2, Problem 60E

a.

To determine

To find: The limit $\underset{x\to 0}{\lim }f\left(x\right)$ .

Answer to Problem 60E

The graph is shown below.

Explanation of Solution

Given:

The given information is the function $f\left(x\right)=\frac{1-\cos x}{x+x^2}$ ,

Calculation:

Here, the graph of the given function is,

  

b.

To determine

To find: The error in the following incorrect application of l’Hopital’s rule.

Answer to Problem 60E

The answer is shown below.

Explanation of Solution

Given:

  $\begin{array}{c}\underset{x\to 0}{\lim }=\frac{1-\cos x}{x+x^2}\\ =\underset{x\to 0}{\lim }\frac{\sin x}{1+2x}\\ =\frac{\cos 2x}{2}\\ =\frac{1}{2}\end{array}$

Calculation:

The error in the following application is,

  $\begin{array}{c}\underset{x\to 0}{\lim }\frac{\sin x}{1+2x}=\frac{0}{1}\\ =0\end{array}$

Since it cannot be equal to $\frac{1}{2}$

Therefore, the error is shown above.