Question

use the following function and its graph to answer parts a through d below. let f(x)= \begin{cases} 5 - x, & x < 3 \\ 3, & x = 3 \\ dfrac{2x}{3}, & x > 3 end{cases} graph shown \\ a. $limlimits_{x \to 3^+} f(x) = 2$ (simplify your answer.) \\ b. the limit does not exist. \\ find $limlimits_{x \to 3^-} f(x)$. select the correct choice below and, if necessary, fill in the answer box in your choice. \\ a. $limlimits_{x \to 3^-} f(x) = square$ (simplify your answer.) \\ b. the limit does not exist.

Explanation:

Step1: Identify the function for \( x \to 3^- \)

For \( x < 3 \), the function is \( f(x)=5 - x \).

Step2: Evaluate the limit as \( x \to 3^- \)

To find \( \lim_{x \to 3^-} f(x) \), we substitute \( x = 3 \) into the function \( 5 - x \) (since the limit as \( x \) approaches 3 from the left depends on the function for \( x < 3 \)).
So, \( \lim_{x \to 3^-} (5 - x)=5 - 3 = 2 \).

Answer:

A. \( \lim\limits_{x \to 3^-} f(x) = \boldsymbol{2} \)