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. ( limlimits_{x \to 2^+} f(x) = 3 ) (simplify your answer.)
b. the limit does not exist.

find ( limlimits_{x \to 2^-} f(x) ). select the correct choice below and, if necessary, fill in the answer box in your choice.

a. ( limlimits_{x \to 2^-} f(x) = square ) (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Determine the applicable function

For \( x \to 2^- \), we have \( x < 2 \), so the function \( f(x)=5 - x \) (since \( 2<3 \), the first piece of the piecewise function applies).

Step2: Evaluate the limit

To find \( \lim_{x \to 2^-} f(x) \), we substitute \( x = 2 \) into \( 5 - x \) (because the function is continuous for \( x < 3 \) in this piece). So \( \lim_{x \to 2^-} (5 - x)=5 - 2 \).

Step3: Calculate the result

\( 5 - 2 = 3 \).

Answer:

\( \lim_{x \to 2^-} f(x)=\boxed{3} \) (so the correct choice is A with the answer 3)