Question: Find all values of real number a such that $ \lim_{x\to1}\frac{ax^2+a^2x-2}{x^3-3x+2} $ exists. Thanks in advance for looking at my question.
I was tackling this limits problem using this method, but I can't seem to find any error with my work.

Question: 
  Find all values of real number a such that
  $$
\lim_{x\to1}\frac{ax^2+a^2x-2}{x^3-3x+2}
$$
  exists.

My Solution: Suppose $\lim_{x\to1}\frac{ax^2+a^2x-2}{x^3-3x+2}$ exists and is equals to $L$.
We have $$\lim_{x\to1}{ax^2+a^2x-2}=\frac{\lim_{x\to1}ax^2+a^2x-2}{\lim_{x\to1}x^3-3x+2}*\lim_{x\to1}x^3-3x+2=L*0=0$$
Therefore, $\lim_{x\to1}{ax^2+a^2x-2}=0$
implying $a(1)^2+a^2(1)-2=0$.
Solving for $a$, we get $a=-2$ or $a=1$. 
Apparently, the answer is only $a=-2$. I understand where they are coming from, but I can't see anything wrong with my solution either. 

Your solution doesn't work because you multiply through by an indeterminate form, i.e. $0/0$. 
