<s>[INST] How can I determine if the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges or diverges? Determine whether the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges. If it converges find the limit and if it diverges determine whether it has an infinite limit.
Proof: let $a_{n} = \frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$
Divide by $n^{2}$  so you have $\frac{1+\frac{3^n}{n^2}}{\frac{2^{n}}{n^{2}}+n^{2}}$
Therefore it converges to 0 ?
Is that right?
 [/INST] your sequence is equivalent to $a_n=\frac{\frac{n^2}{2^n}+\left(\frac{3}{2}\right)^n}{1+\frac{n^4}{2^n }}$ this goes to infinity for $n$ tends to infinity
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