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.Q:

Computing the limit of a sum: evaluating $\lim_{n \to \infty} \frac{n^3+9n^2+22n+33}{2n^3+12n^2+54n+31}$

The problem
Compute the limit of:
$$\lim_{n \to \infty} \dfrac{n^3+9n^2+22n+33}{2n^3+12n^2+54n+31}$$

My attempt
The numerator and denominator both have polynomial growth. From Lagrange multiplier form I can use Taylor series to evaluate the limit to $$1$$
My question
Is there a more efficient way to do this?
I tried to use the ratio test but I got the wrong answer

A:

Note that:
$$\begin{align}\lim_{n \to \infty} \dfrac{n^3+9n^2+22n+33}{2n^3+12n^2+54n+31}&= \lim_{n \to \infty} \dfrac{n^3+9n^2+22n+33}{2(n+1)(n+2)(n+3)}\\&=\lim_{n \to \infty} \dfrac{n(n+1)(n+2)(n+3)+n^3+9n^2+22n+33}{2(n+1)(n+2)(n+3)}\\&=\lim_{n \to \infty} \dfrac{n(n+1)(n+2)(n+3)+\frac{1}{3}n(n+1)(n+2)(n+3)+O\left(\frac{1}{n}\right)}{2(n+1)(n+2)(n+3)}\\&=\lim_{n \to \infty} \dfrac{3(n+1)(n+2)(n+3)}{2(n+1)(n+2)(n+

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