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If $ f(x) = \left\{ \begin{array}{ll} \ln x & \quad 0 \lt x \leq 2 \\\ x^2 \ln 2 & \quad 2 \lt x \leq 4 \end{array} \right. $ then, $\lim \limits_{x \to 2} f(x)$ is:

A

$\ln 2$

B

$\ln 8$

C

$\ln 16$

D

The limit does not exist.

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