Let's evaluate the limit [tex]\(\lim_{x \rightarrow 7} f(x)\)[/tex] for the given piecewise function:
[tex]\[
f(x) = \begin{cases}
15 & \text{if } x < -4 \\
6 - 2x & \text{if } x \geq -4
\end{cases}
\][/tex]
1. Understand the behavior of the function at [tex]\(x = 7\)[/tex]:
Given that [tex]\(x \to 7\)[/tex], we should identify which piece of the piecewise function applies when [tex]\(x\)[/tex] is 7. By examining the conditions:
- For [tex]\(x < -4\)[/tex], [tex]\(f(x) = 15\)[/tex]
- For [tex]\(x \geq -4\)[/tex], [tex]\(f(x) = 6 - 2x\)[/tex]
Since [tex]\(7 \geq -4\)[/tex], we use the second piece of the function [tex]\(6 - 2x\)[/tex].
2. Substitute [tex]\(x = 7\)[/tex] into the function [tex]\(6 - 2x\)[/tex]:
[tex]\[
f(7) = 6 - 2(7)
\][/tex]
3. Perform the arithmetic:
[tex]\[
f(7) = 6 - 2 \cdot 7 = 6 - 14 = -8
\][/tex]
Thus, the limit as [tex]\(x\)[/tex] approaches 7 of the function [tex]\(f(x)\)[/tex] is:
[tex]\[
\lim_{x \rightarrow 7} f(x) = -8
\][/tex]
