(d) \( \lim _{x \rightarrow 2}\left\{\begin{array}{ll}1-x & x \leq 2 \\ 3 x^{2}+1 & x>2\end{array}\right. \)

To find the limit \( \lim _{x \rightarrow 2}\left\{\begin{array}{ll}1-x & x \leq 2 \\ 3 x^{2}+1 & x>2\end{array}\right. \), we can analyze both pieces of the piecewise function as \( x \) approaches 2. 1. For \( x \leq 2 \), we use the function \( 1-x \): \[ \lim_{x \rightarrow 2^-} (1-x) = 1 - 2 = -1 \] 2. For \( x > 2 \), we use the function \( 3x^2 + 1 \): \[ \lim_{x \rightarrow 2^+} (3x^2 + 1) = 3(2^2) + 1 = 3 \times 4 + 1 = 12 + 1 = 13 \] Since the limit from the left (as \( x \) approaches 2 from the left) is \(-1\) and the limit from the right (as \( x \) approaches 2 from the right) is \(13\), the limits do not match. Therefore, the overall limit does not exist. Thus, we conclude: \[ \lim _{x \rightarrow 2} \left\{\begin{array}{ll}1-x & x \leq 2 \\ 3 x^{2}+1 & x>2\end{array}\right. \text{ does not exist (DNE)}. \]