Question

In Exercises \(67 - 70\) , use the following function. \(f ( x ) = \left\\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.\) Multiple Choice What is the value of \(\lim _ { x \rightarrow 1 ^ { - } } f ( x ) ?\) \(\begin{array} { l l l l l } { \text { (A) } 5 / 2 } & { \text { (B) } 3 / 2 } & { \text { (C) } 1 } & { \text { (D) } 0 } & { \text { (E) does not exist } } \end{array}\)

Short answer

The value of \( \lim _ { x \rightarrow 1 ^ { - } } f ( x ) \) is \( 1 \), which corresponds to option (C) in the multiple-choice options.

Step-by-step solution

Identifying the relevant section of the function

The task is to find the left-hand limit as x approaches 1. This means we will use the part of the piece-wise function defined for \( x \leq 1 \). In this case, that part is \( 2 - x \).

Substitute the limit value

Substitute \( x = 1 \) into this function to get \( 2 - 1 \).

Calculate the limit

Calculate the result to give a final answer of \( 1 \).