Chapter 12.1, Problem 69E

(a)

To determine

To find: The conclusion about $\underset{x\to 2}{\lim }f\left(x\right)$ is exist or not if $f\left(2\right)=4$ .

Answer to Problem 69E

The $\underset{x\to a}{\lim }f\left(x\right)$ may not exists even the value of the function is defined at the point $a$ .

Explanation of Solution

Consider the following example of function:

  $\begin{array}{c}f\left(x\right)=xx\ne 2\\ =4x=2\end{array}$

By the above example it is clear that $f\left(2\right)=4$ but the limit does not exists.

So we can conclude that even if the value of function is given at a point say $a$ that does not implies that limit of the function at that point $a$ also exists.

(b)

To determine

To find: The conclusion about $f\left(2\right)$ if $\underset{x\to 2}{\lim }f\left(x\right)=4$ .

Answer to Problem 69E

The conclusion is that the value of $f\left(2\right)=4$ .

Explanation of Solution

If $\underset{x\to 2}{\lim }f\left(x\right)=4$ is given it means the limit of the function at the point $2$ is exists and hence $\underset{x\to 2}{\lim }f\left(x\right)=f\left(2\right)$ .

So the conclusion is that the value of $f\left(2\right)=4$ .