Math  /  Calculus

QuestionGiven the function f(x)={2x+18 if x<6x+42 if x>62 if x=6f(x)=\left\{\begin{array}{lll}2 x+18 & \text { if } & x<-6 \\ \sqrt{x+42} & \text { if } & x>-6 \\ 2 & \text { if } & x=-6\end{array}\right., determine the truth of the following statements about f(6)f(-6) and its limits.

Studdy Solution
The function f(x)f(x) is continuous at x=6x=-6 if and only if the limit of f(x)f(x) as xx approaches 6-6 equals f(6)f(-6). Since f(6)=2f(-6) =2 and limx6f(x)=6\lim{{x \to -6}} f(x) =6, the function is not continuous at x=6x=-6.
So, the statements we agree with are1. f(6)f(-6) is defined.
2. limx6f(x)\lim{x \rightarrow-6} f(x) exists.
3. The function is not continuous at x=6x=-6.

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