Chapter 11.1, Problem 79E

(a)

To determine

To conclude:about the $\underset{x\to 2}{\lim }f\left(x\right)$

Answer to Problem 79E

Here, nothing is concluded about $\underset{x\to 2}{\lim }f\left(x\right)$ .

Explanation of Solution

Given: $f\left(2\right)=4$ and $\underset{x\to 2}{\lim }f\left(x\right)=4$

Calculation:

Even if it is known that $f\left(2\right)=4$ , nothing is concluded about $\underset{x\to 2}{\lim }f\left(x\right)$ because the function may not approach 4 as $x$ approaches 2 from either the positive or negative direction or both.

Conclusion:

Hence,nothing is concluded about $\underset{x\to 2}{\lim }f\left(x\right)$ .

(b)

To determine

To conclude: about the $f\left(2\right)$

Answer to Problem 79E

Here, nothing is concluded about $f\left(2\right)$ .

Explanation of Solution

Given: $\underset{x\to 2}{\lim }f\left(x\right)=4$

Calculation:

Only knowing that $\underset{x\to 2}{\lim }f\left(x\right)=4$ , nothing is concluded about $f\left(2\right)$ , since although the function may take values increasingly close to 4 as $x$ approaches 2 from both the positive and negative directions, the function itself may not be defined for $x=2$ , or may take a different value than 4 there.

Conclusion:

Hence, nothing is concluded about $f\left(2\right)$ .