Chapter 11.1, Problem 40E

To determine

To explain:

The limit of $\underset{x\to 1}{\lim }\cos \left(\pi x\right)$ if exists and in case not exist; give reason.

Answer to Problem 40E

The limit of $\underset{x\to 1}{\lim }\cos \left(\pi x\right)$ is $-1$

Explanation of Solution

Given information:

  $\underset{x\to 1}{\lim }\cos \left(\pi x\right)$

  

Firstly determine limit at specified point. Then check the limit exists at the specified point by the points.

Some conditions for existence of the limits:

When the function satisfies the conditions, then limit of function doesn’t exist.

1. $f\left(x\right)$ increases or decreases without a bound as $x$ approaches to $c$ .
2. $f\left(x\right)$ approaches the different number from right side of the $c$ than its approach from left side of the $c$ .
3. $f\left(x\right)$ oscillates between $2$ fixed values as $x$ approaches to $c$ .

From this graph, the function value approaches same values as $x$ approaches from right and left as:

  $\begin{array}{l}\underset{x\to +1}{\lim }\cos \left(\pi x\right)=\cos \left(\pi \right)=-1\\ \underset{x\to -1}{\lim }\cos \left(\pi x\right)=\cos \left(-\pi \right)=-1\end{array}$

Therefore, the limit of the function $\underset{x\to 1}{\lim }\cos \left(\pi x\right)$ is $-1$ .