Let [tex]$f(x)=\left\{\begin{array}{ll}x^2+5x+4, & \text{if } x\ \textless \ 6 \\ 24, & \text{if } x=6 \\ -5x+1, & \text{if } x\ \textgreater \ 6\end{array}\right.$]

Evaluate the following:
(a) [tex]\lim _{x \rightarrow 6^{-}} f(x)=[/tex] [tex]\square[/tex]
(b) [tex]\lim _{x \rightarrow 6^{+}} f(x)=[/tex] [tex]\square[/tex]

(c) Which of the following is true about [tex]f(x)[/tex] at [tex]x=6[/tex]?
- The function has a jump discontinuity



Answer :

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].