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.