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} ) a. find ( limlimits_{x \to 3^+} f(x) ). select the correct choice below and, if necessary, fill in the answer box in your choice. ( \boldsymbol{odot} ) a. ( limlimits_{x \to 3^+} f(x) = square ) (simplify your answer.) ( \boldsymbol{circ} ) b. the limit does not exist.

Explanation:

Step1: Identify the right-hand limit function

For \( x \to 3^+ \), we use the part of the piecewise function where \( x > 3 \), which is \( f(x)=\frac{2x}{3} \).

Step2: Substitute \( x = 3 \) into the function

To find the limit as \( x \) approaches 3 from the right, we substitute \( x = 3 \) into \( \frac{2x}{3} \). So we have \( \frac{2\times3}{3} \).

Step3: Simplify the expression

Simplifying \( \frac{2\times3}{3} \), the 3 in the numerator and denominator cancels out, leaving us with 2.

Answer:

\( \lim\limits_{x\to 3^+} f(x) = 2 \)