Question

for the function ( f(x) ) shown below, determine ( limlimits_{x \to 4^+} f(x) ). ( f(x) = \begin{cases} 9 - 3x & \text{for } x geq 4 \\ x^2 - 10 & \text{for } x < 4 end{cases} )

Explanation:

Step1: Identify the relevant piece of the function

To find the right - hand limit as \(x
ightarrow4^{+}\), we look at the part of the piece - wise function where \(x\geq4\) (since \(x
ightarrow4^{+}\) means \(x\) approaches 4 from values greater than 4). The function for \(x\geq4\) is \(f(x)=9 - 3x\).

Step2: Substitute \(x = 4\) into the relevant function

We substitute \(x = 4\) into \(f(x)=9 - 3x\) to find the limit. Using the formula for the limit of a polynomial function (the limit of a polynomial \(a_nx^n+\cdots+a_1x + a_0\) as \(x
ightarrow c\) is \(a_nc^n+\cdots+a_1c + a_0\)), we have:
\(\lim_{x
ightarrow4^{+}}f(x)=\lim_{x
ightarrow4^{+}}(9 - 3x)\)
Substitute \(x = 4\) into \(9-3x\): \(9-3\times4=9 - 12=- 3\)

Answer:

\(-3\)