Question

If limx→3f(x)=7, which of the following must be true? I. f is continuous at x = 3 II. f is differentiable at x = 3

Answer 1

Without knowing anything else about $f(x)$, neither of these need be true.

Suppose

$f(x)=|x-3|+7=\begin{cases}10-x&\text{for }x3\\0&\text{for }x=3\end{cases}$

Then (I) isn't true because, while the limit exists as $x\to3$ and is equal to 7, we have $f(3)=0\neq7$, so $f(x)$ is not continuous there.

(II) also true because $f(x)$ is not differentiable at $x=3$; that is,

$\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3}(10-x)'=\lim_{x\to3}(-1)=-1$

but

$\displaystyle\lim_{x\to3^+}f'(x)=\lim_{x\to3}(x+4)'=\lim_{x\to3}1=1$

which means the derivative does not exist at $x=3$.