Let's carefully evaluate the given piecewise function step-by-step to find the required limits and analyze the continuity at [tex]\( x = 6 \)[/tex].
Given function:
[tex]\[
f(x)=
\begin{cases}
x^2 + 5x + 4, & \text{if } x < 6 \\
24, & \text{if } x = 6 \\
-5x + 1, & \text{if } x > 6
\end{cases}
\][/tex]
(a) To find [tex]\(\lim_{x \rightarrow 6^{-}} f(x)\)[/tex], we need to look at the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 6 from the left:
[tex]\[ f(x) = x^2 + 5x + 4 \text{ for } x < 6 \][/tex]
So, we evaluate the limit:
[tex]\[ \lim_{x \rightarrow 6^{-}} (x^2 + 5x + 4) = 6^2 + 5(6) + 4 = 36 + 30 + 4 = 70 \][/tex]
Thus,
[tex]\[ \lim_{x \rightarrow 6^{-}} f(x) = 69.9999999983 \][/tex]
(b) To find [tex]\(\lim_{x \rightarrow 6^{+}} f(x)\)[/tex], we need to look at the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 6 from the right:
[tex]\[ f(x) = -5x + 1 \text{ for } x > 6 \][/tex]
So, we evaluate the limit:
[tex]\[ \lim_{x \rightarrow 6^{+}} (-5x + 1) = -5(6) + 1 = -30 + 1 = -29 \][/tex]
Thus,
[tex]\[ \lim_{x \rightarrow 6^{+}} f(x) = -29.000000000500002 \][/tex]
(c) In order to determine the continuity of [tex]\( f(x) \)[/tex] at [tex]\( x = 6 \)[/tex], we need to compare the left-hand limit and the right-hand limit at [tex]\( x = 6 \)[/tex] as well as the value of the function at [tex]\( x = 6 \)[/tex]:
- We have [tex]\(\lim_{x \rightarrow 6^{-}} f(x) = 69.9999999983 \)[/tex]
- We have [tex]\(\lim_{x \rightarrow 6^{+}} f(x) = -29.000000000500002 \)[/tex]
- [tex]\( f(6) = 24 \)[/tex]
Since the left-hand limit ([tex]\( 69.9999999983 \)[/tex]) and right-hand limit ([tex]\( -29.000000000500002 \)[/tex]) are not equal, the function [tex]\( f(x) \)[/tex] has a jump discontinuity at [tex]\( x = 6 \)[/tex].
In summary:
(a) [tex]\( \lim_{x \rightarrow 6^{-}} f(x) = 69.9999999983 \)[/tex]
(b) [tex]\( \lim_{x \rightarrow 6^{+}} f(x) = -29.000000000500002 \)[/tex]
(c) The function [tex]\( f(x) \)[/tex] has a jump discontinuity at [tex]\( x = 6 \)[/tex].
