For the function $f(x)=|3 x-9|$ , evaluate the left and right limits of $f^{\prime}(x)$ as $x$ approaches $2$ . $\newline$ a. $\lim _{x \rightarrow 2^{-}} f(x) \quad 3$ $\newline$ b. $\lim _{x \rightarrow 2^{+}} f(x) \quad 3$

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Q. For the function

f(x)=|3 x-9|

, evaluate the left and right limits of

f^{\prime}(x)

x

approaches

2

\newline

\lim _{x \rightarrow 2^{-}} f(x) \quad 3

\newline

\lim _{x \rightarrow 2^{+}} f(x) \quad 3

Understand Absolute Value Function: To find the left and right limits of $f'(x)$ as $x$ approaches $2$ , we first need to understand the behavior of the absolute value function $f(x)=|3x-9|$ . The absolute value function has a kink at the point where the expression inside the absolute value is zero. In this case, that point is when $3x-9=0$ , which occurs at $x=3$ . Since we are interested in the limits as $x$ approaches $2$ , we are looking at the behavior of the function to the left and right of $x=2$ , but still to the left of the kink at $x=3$ .
Analyze Function Behavior: For $x < 3$ , the expression inside the absolute value, $3x-9$ , is negative. Therefore, the function $f(x)$ for $x < 3$ is $f(x) = -(3x-9) = -3x + 9$ . The derivative of this function with respect to $x$ is $f'(x) = -3$ .
Calculate Derivative: For $x > 3$ , the expression inside the absolute value, $3x-9$ , is positive. Therefore, the function $f(x)$ for $x > 3$ is $f(x) = 3x-9$ . The derivative of this function with respect to $x$ is $f'(x) = 3$ .
Determine Left and Right Limits: Since we are interested in the limits as $x$ approaches $2$ , we only need to consider the behavior of the function for $x < 3$ . Therefore, both the left-hand limit and the right-hand limit of $f'(x)$ as $x$ approaches $2$ are equal to the derivative of the function for $x < 3$ , which is $-3$ .
Calculate Left-Hand Limit: The left-hand limit of $f'(x)$ as $x$ approaches $2$ is: $\lim_{x \to 2^{-}} f'(x) = -3$
Calculate Right-Hand Limit: The right-hand limit of $f'(x)$ as $x$ approaches $2$ is: $\lim_{x \to 2^{+}} f'(x) = -3$