Let f(x) be a continuous function in R. If the limit $\displaystyle \lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2}$ exists and it is a real number, find the value of $\displaystyle \alpha$ so that the function g(x) = $\displaystyle \lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2} \quad for \quad x \ne 2$

$\displaystyle \alpha \quad for \quad x = 2 $

is continuous

I am trying to prove that g(x) is differentiable, because when a function is differentiable is continuous as well, but I can't. Can anyone help?