Rearranging Limit Problem Having some trouble expressing the following:
Find $\displaystyle \lim_{x\to\infty} \frac{2-3x+4x^2}{7 + 12x + 3x^2}$, by first expressing the limit as $\displaystyle \lim_{x\to0} f(x)$, for some real function $\displaystyle f$
My initial thoughts would be to switch the numerator and the denominator and then evaluate at $\displaystyle x\to0$, which would leave an answer of $\displaystyle \frac{7}{2}$, but this seems too easy and is thus more than likely incorrect.

First write
$$
{2-3x+4x^2\over 7+12x+3x^2}= 
{{2\over x^2}-{3\over x}+4\over {7\over x^2}+{12\over x}+3}.
$$
Then the limit is expressed by (setting $u=1/x$ in the above)
$$
\lim_{u\rightarrow0}{{{2u^2}-{3u}+4\over {7u^2}+{12u}+3}} .
$$
