For which values of $a$ does $\lim\limits _{x\to a}\frac{|x^2-(2+a)x+2a|}{x-a}$ exist?

Question

For which values of $a\in{\mathbb{R}}$ the limit $$\lim_{x\to{a}}{}{\frac{|x^2-(2+a)x+2a|}{x-a}}$$ exists?

I suspect that I need to remove the absolute value and then factor the numerator and last divide with $x-a$ in order to remove the undefined form of $\frac{0}{0}$, but am I correct?

Answer 1

Hint. Note that $$x^2-(2+a)x+2a=(x-2)(x-a).$$ Moreover, pay attention to the fact that $|x-a|/(x-a)=1$ if $x>a$ and it is equal to $-1$ for $x<a$.