Question

use the following function and its graph to answer parts a through d below. let ( f(x) = \begin{cases} 8 - x, & x < 6 \\ 3, & x = 6 \\ \frac{x}{3}, & x > 6 end{cases} ) a. find ( lim_{x \to 6^+} f(x) ). select the correct choice below and, if necessary, fill in the answer box in your choice. a. ( lim_{x \to 6^+} f(x) = 2 ) (simplify your answer.) b. the limit does not exist. find ( lim_{x \to 6^-} f(x) ). select the correct choice below and, if necessary, fill in the answer box in your choice. a. ( lim_{x \to 6^-} f(x) = square ) (simplify your answer.)

Explanation:

Step1: Identify the function for \( x \to 6^- \)

For \( x < 6 \), the function is \( f(x) = 8 - x \).

Step2: Substitute \( x = 6 \) into the function

To find the left - hand limit as \( x \to 6^- \), we use the function \( f(x)=8 - x \) (since for values less than 6, this is the relevant part of the piece - wise function). We substitute \( x = 6 \) into \( 8 - x \): \( 8-6=2 \).

Answer:

\( \lim\limits_{x\to 6^-}f(x)=2 \)