Chapter 11.1, Problem 46E

To determine

Graph the function $f\left(x\right)=\cos \frac{1}{x},\underset{x\to 0}{\lim }f\left(x\right)$ and determine whether the limit exists

Answer to Problem 46E

The limit does not exist.

Explanation of Solution

Given:

  $f\left(x\right)=\cos \frac{1}{x},\underset{x\to 0}{\lim }f\left(x\right)$

Graph:

By using a graphing utility, the graph of given function is

  

Interpretation:

The graph shows that as $x$ approaches 0 from the left, the limit is -0.11and as $x$ approaches 0 from the right, the limit is -0.922.

The limit does not exist because $\underset{x\to a-}{\lim }f\left(x\right)\ne \underset{x\to a+}{\lim }f\left(x\right)$ . The sum diverges if the limit of $f\left(x\right)$ is not zero or does not exist. As x approaches 0, $\cos \frac{1}{x}$ does not attain a limit but rather it oscillates.

Conclusion:

Therefore, the limit does not exist as $x$ approaches 0.