Let $$ f(x)=\left\{\begin{array}{ll}x^{2}, & ext { for } x2\end{array} ight. $$. Find each limit. $$ \begin{array}{ll} ext { (A), } \lim _{x ightarrow 2^{+}} f(x) & ext { (B) } \lim _{x ightarrow 2^{-}} f(x) \ ext { (C) } \lim _{x ightarrow 2} f(x) & ext { (D) } f(2)\end{array} $$ $$ \begin{array}{l} ext { A. Select the correct choice and, if necessary, fill in the answer box to complete your choice. } \ ext { (A) }(x)=\square ext { (Type an integer or a decimal.) } \ ext { B. The limit does not exist. }\end{array} $$

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Let $$ f(x)=\left\{\begin{array}{ll}x^{2}, & 	ext { for } x<2 \ 3 x, & 	ext { for } x>2\end{array}ight. $$. Find each limit. $$ \begin{array}{ll}	ext { (A), } \lim _{x ightarrow 2^{+}} f(x) & 	ext { (B) } \lim _{x ightarrow 2^{-}} f(x) \ 	ext { (C) } \lim _{x ightarrow 2} f(x) & 	ext { (D) } f(2)\end{array} $$ $$ \begin{array}{l}	ext { A. Select the correct choice and, if necessary, fill in the answer box to complete your choice. } \ 	ext { (A) }(x)=\square 	ext { (Type an integer or a decimal.) } \ 	ext { B. The limit does not exist. }\end{array} $$

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To find the limits and the value of the function $$ f(x) $$ at $$ x = 2 $$, we will evaluate each limit step by step. ### Given Function The function is defined as: $$ f(x) = \begin{cases} x^2 & \text{for } x < 2 \ 3x & \text{for } x > 2 \end{cases} $$ ### (A) $$ \lim_{x \rightarrow 2^{+}} f(x) $$ To find the limit as $$ x $$ approaches 2 from the right ($$ 2^{+} $$), we use the definition of $$ f(x) $$ for $$ x > 2 $$: $$ f(x) = 3x $$ Thus, $$ \lim_{x \rightarrow 2^{+}} f(x) = 3(2) = 6 $$ ### (B) $$ \lim_{x \rightarrow 2^{-}} f(x) $$ To find the limit as $$ x $$ approaches 2 from the left ($$ 2^{-} $$), we use the definition of $$ f(x) $$ for $$ x < 2 $$: $$ f(x) = x^2 $$ Thus, $$ \lim_{x \rightarrow 2^{-}} f(x) = 2^2 = 4 $$ ### (C) $$ \lim_{x \rightarrow 2} f(x) $$ The limit as $$ x $$ approaches 2 exists only if both one-sided limits are equal. We have: - $$ \lim_{x \rightarrow 2^{+}} f(x) = 6 $$ - $$ \lim_{x \rightarrow 2^{-}} f(x) = 4 $$ Since these two limits are not equal, the limit does not exist: $$ \lim_{x \rightarrow 2} f(x) \text{ does not exist.} $$ ### (D) $$ f(2) $$ The function $$ f(x) $$ is not defined at $$ x = 2 $$ since there is no case for $$ x = 2 $$ in the piecewise definition. Therefore: $$ f(2) \text{ is undefined.} $$ ### Summary of Results - (A) $$ \lim_{x \rightarrow 2^{+}} f(x) = 6 $$ - (B) $$ \lim_{x \rightarrow 2^{-}} f(x) = 4 $$ - (C) $$ \lim_{x \rightarrow 2} f(x) \text{ does not exist.} $$ - (D) $$ f(2) \text{ is undefined.} $$ ### Final Answers - (A) $$ 6 $$ - (B) The limit does not exist. - (C) The limit does not exist. - (D) Undefined. - (A) 6 - (B) The limit does not exist. - (C) The limit does not exist. - (D) Undefined.

Let . Find each limit.

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