<s>[INST] How to calculate $\lim_{x\to\infty}\frac{7x^4+x^2 3^x+2}{x^3+x 4^x+1}$? 
$$\lim_{x\to\infty}\frac{7x^4+x^2 3^x+2}{x^3+x 4^x+1}$$ 

I can't  seem to find away to get rid of the $3^x$ and $4^x$ and then resolve it.
 [/INST] Use equivalents:
$$\begin{aligned}
7x^4+x^2\, 3^x+2&\sim_\infty x^2 3^x\\
x^3+x\,4^x+1&\sim_\infty x\,4^x
\end{aligned}\quad\text{hence}\quad\frac{7x^4+x^2\, 3^x+2}{x^3+x\,4^x+1}\sim_\infty\frac{ x^2 3^x}{ x\,4^x}=x\Bigl(\frac34\Bigr)^{\!x}\to 0.$$
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