Chapter 1.1, Problem 75E

To determine

To choose: The correct option.

Answer to Problem 75E

The correct option is (D) Does not exists

Explanation of Solution

Given information: The limit $\underset{x\to 1}{\lim }f(x)$

Calculation:

Consider the given limit:

  $\underset{x\to 1}{\lim }f(x)$

From (73) of this solution, know that $\underset{x\to 1^{-}}{\lim }f(x)=1$

From (74) of this solution, know that $\underset{x\to 1^{+}}{\lim }f(x)=\frac{3}{2}$

The two one-sided limits must be equal in order for the two-sided limit to exist, according to Theorem 3.

The requested/given two-sided limit at $x=1$ does not exist since the two one-sided limits are not equal.

Therefore the answer is $\underset{x\to 3}{\lim }f(x)=\text{ does not exist }$

Hence the required correct option is (D) Does not exists.