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

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.