Full solution

2

. For the function

f(x)=|3 x-9|

, evaluate the left and right limits of

f(x)

x

approaches

2

\newline

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

\newline

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

Consider Values Less Than $2$ : To find the left-hand limit as $x$ approaches $2$ , we need to consider the values of $x$ that are just less than $2$ . For these values, the expression inside the absolute value, $3x - 9$ , will be negative because $3(2) - 9 = 6 - 9 = -3$ . Therefore, the absolute value function will negate the inside expression to make it positive.
Write Left-Hand Limit Expression: We can write the expression for the left-hand limit as: $\newline$ $\lim_{x \rightarrow 2^{-}}f(x) = \lim_{x \rightarrow 2^{-}}|3x - 9|$ $\newline$ Since $3x - 9$ is negative when $x$ is just less than $2$ , we can remove the absolute value by negating the expression inside: $\newline$ $\lim_{x \rightarrow 2^{-}}f(x) = \lim_{x \rightarrow 2^{-}}-(3x - 9)$
Evaluate Left-Hand Limit: Now we can evaluate the limit by substituting $x$ with a value just less than $2$ : $\lim_{x \rightarrow 2^{-}}f(x) = -(3(2) - 9) = -(6 - 9) = -(-3) = 3$
Consider Values Greater Than $2$ : To find the right-hand limit as $x$ approaches $2$ , we need to consider the values of $x$ that are just greater than $2$ . For these values, the expression inside the absolute value, $3x - 9$ , will be negative or positive depending on whether $x$ is closer to $2$ or further away. However, since we are considering values just greater than $2$ , the expression $3x - 9$ will still be negative because $3(2) - 9 = 6 - 9 = -3$ . Therefore, the absolute value function will negate the inside expression to make it positive.
Write Right-Hand Limit Expression: We can write the expression for the right-hand limit as: $\newline$ $\lim_{x \rightarrow 2^{+}}f(x) = \lim_{x \rightarrow 2^{+}}|3x - 9|$ $\newline$ Since $3x - 9$ is negative when $x$ is just greater than $2$ , we can remove the absolute value by negating the expression inside: $\newline$ $\lim_{x \rightarrow 2^{+}}f(x) = \lim_{x \rightarrow 2^{+}}-(3x - 9)$
Evaluate Right-Hand Limit: Now we can evaluate the limit by substituting $x$ with a value just greater than $2$ : $\lim_{x \rightarrow 2^{+}}f(x) = -(3(2) - 9) = -(6 - 9) = -(-3) = 3$