Chapter 12.1, Problem 31E

To determine

To find: the limit, if it exists.

Answer to Problem 31E

The limit of the function $f\left(x\right)=2\cos \frac{\pi }{x}$ does not exist at the specified point

Explanation of Solution

Given information:

Given figure

  

Calculation:

Consider the following of the function $f\left(x\right)=2\cos \frac{\pi }{x}$

  

Check whether the limit exists at the specified point by verifying the following points.

Conditions for Existence of limits:

When a function $f\left(x\right)$ satisfies any of the following conditions, then the limit of the function does not exist

1. $f\left(x\right)$ Approaches a different number from the right side of $c$ than it approaches $c$ from the left.
2. $f\left(x\right)$ Increases or decreases without bound as $x$ Approaches $c$ .
3. $f\left(x\right)$ Oscillates between two fixed values as $x$ Approaches $c$ .

On checking for the three conditions

The graph satisfies the third condition.

As the function $f\left(x\right)$ oscillates between two fixed values as $x$ approaches 0 from either side, limit of the function at $x=0$ is not defined.

Hence limit of the function $f\left(x\right)=2\cos \frac{\pi }{x}$ does not exist at the specified point.