I don't understand why the limit of the polynomial is incorrect. $$\lim_{x\to\infty}a_1x^n+a_2x^{n-1}+...=\lim_{x\to\infty}a_1x^n(1+\frac{a_2}{a_1}x^{-1}+...)=\lim_{x\to\infty}a_1x^n\cdot\lim_{x\to\infty}(1+\frac{a_2}{a_1}x^{-1}+...)=\lim_{x\to\infty}a_1x^n$$ I originally thought this was right until it failed under some circumstances. I think it is the $\infty$ that makes the statement false. Please tell me which step is illegal and why is it illegal.
$a_1\ne0$ and polynomial of degree n
Everything you did in the question body is ok. The problem from the comments is here:
$$\lim_{x\to\infty}1=\color{red}{\lim_{x\to\infty}(a_1x^n-a_1x^n+1)=\lim_{x\to\infty}(a_1x^n+1)-\lim_{x\to\infty}a_1x^n}$$
we can't use the limit addition rule to split into terms whose limits are not finite. ($\infty-\infty$ is an indeterminate form).