Basic Limit Properties Question 2

If $f(x)=\begin{cases} \ln x & \text{for } 0\lt x \leq 2 \\ x^2\ln 2 & \text{for } 2\lt x\leq 4 \end{cases}$ , then $\lim\limits_{x\to 2}f(x)$ is

\ln2

\ln8

\ln16

nonexistent

Approach

Find the left and right hand limits.

Left-hand Limit

\lim\limits_{x\to 2^-}f(x) = \ln(2)

= \ln 2

Right-hand Limit

\lim\limits_{x\to 2^+}f(x) = (2)^2\ln(2)

= 4\ln2

Since the left and right hand limits do not equal each other, the limit is nonexistent.