<s>[INST] Best way to evaluate $\lim_{n \rightarrow \infty} |\frac{(3(n+1)+4)(4^{n+1}+5)(5^n+3)}{(5^{n+1}+3)(3n+4)(4^n+5)}|$ Best way to evaluate $\lim_{n \rightarrow \infty} |\frac{(3(n+1)+4)(4^{n+1}+5)(5^n+3)}{(5^{n+1}+3)(3n+4)(4^n+5)}|$
Can you all show me some different ways to evaluate this limit? I was thinking of multiplying the numerator and denominator by something.. maybe $\frac{1}{5^{n+1}}$... I don't know. I'm sure there are a lot of ways to evalute this and you guys will show me a lot of slick awesome ways!!
As you may have guessed, i'm in the midst of employing the ratio test for a series.
 [/INST] We need to factor out the leading terms to obtain
$$\frac{(3(n+1)+4)(4^{n+1}+5)(5^n+3)}{(5^{n+1}+3)(3n+4)(4^n+5)}
=\frac{(n+1)\cdot 4^{n+1} \cdot 5^n}{5^{n+1} \cdot n \cdot 4^n}\frac{\left(3+\frac4{n+1}\right)\left(1+\frac5{4^{n+1}}\right)\left(1+\frac3{5^n}\right)}{\left(1+\frac3{5^{n+1}}\right)\left(3+\frac4n\right)\left(1+\frac 5 {4^n}\right)}$$
with
$$\frac{(n+1)\cdot 4^{n+1} \cdot 5^n}{5^{n+1} \cdot n \cdot 4^n}=\frac{4}{5}\frac{n+1}n \to \frac45$$
and
$$\frac{\left(3+\frac4{n+1}\right)\left(1+\frac5{4^{n+1}}\right)\left(1+\frac3{5^n}\right)}{\left(1+\frac3{5^{n+1}}\right)\left(3+\frac4n\right)\left(1+\frac 5 {4^n}\right)} \to1$$
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