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 \\ dfrac{x}{3}, & x > 6 end{cases} ). a. find ( limlimits_{x \to 6^+} f(x) ). select the correct choice below and, if necessary, fill in the answer box in your choice. ( \bigcirc ) a. ( limlimits_{x \to 6^+} f(x) = ) (simplify your answer.) ( \bigcirc ) b. the limit does not exist.

Explanation:

Step1: Identify the relevant piece of the function

For \( x \to 6^+ \), we use the part of the piecewise function where \( x > 6 \), which is \( f(x)=\frac{x}{3} \).

Step2: Evaluate the limit

Substitute \( x = 6 \) into \( \frac{x}{3} \) (since we're taking the limit as \( x \) approaches 6 from the right, we use the expression for \( x > 6 \)). So, \( \lim_{x \to 6^+} \frac{x}{3}=\frac{6}{3} \).

Step3: Simplify the result

\( \frac{6}{3}=2 \).

Answer:

A. \( \lim\limits_{x \to 6^+} f(x) = 2 \)