Chapter 2.2, Problem 1E

To determine

To explain: The meaning of $\underset{x\to 2}{\lim }f\left(x\right)=5$ and check whether this statement is true for $f\left(2\right)=3$ or not.

Answer to Problem 1E

The meaning of $\underset{x\to 2}{\lim }f\left(x\right)=5$ is, as x approaches 2, f (x) gets closer and closer to 5.

The statement is true.

Explanation of Solution

Suppose the function $f\left(x\right)$ is defined for all $x$ near $2$ but not exactly at $x=2$.

The $\underset{x\to 2}{\lim }f\left(x\right)=5$ exists if $f\left(x\right)$ gets arbitrarily close to $5$ as $x$ gets sufficiently closer (but not equal) to $2$ from both sides of $2$.

Therefore, the limit of the function $f\left(x\right)$ exists and equal to $5$ when x approaches 2 and it can be written as $\underset{x\to 2}{\lim }f\left(x\right)=5$.

Possibly this statement is true for $f\left(2\right)=3$ as the limit depends upon the functional values near to 2 but not exactly at 2.