<s>[INST] taking the limit $\lim\limits_{n\rightarrow \infty} {\frac{(3^{n+1} + 4)(7^n-47)}{(7^{n+1}-47)(3^n +4)} }$ I need help with a guide on how i will deal with this kind of problem.. This a part of my solution in series convergence. I find it hard taking the limit of this: $$\lim_{n\rightarrow \infty} {\frac{(3^{n+1} + 4)(7^n-47)}{(7^{n+1}-47)(3^n +4)} }$$
 [/INST] There are a couple of properties of limits you can abuse here. The one that should pop out first is $\lim f(x)g(x) = \lim f(x) \cdot \lim g(x)$.
$\lim_{n\rightarrow \infty} {\frac{(3^{n+1} + 4)(7^n-47)}{(7^{n+1}-47)(3^n +4)}} = \lim_{n\rightarrow \infty} {\frac{3^{n+1}+4}{3^n+4}} \cdot \lim_{n\rightarrow \infty} {\frac{7^n-47}{7^{n+1}-47}} = 3 \cdot \frac{1}{7} = \frac{3}{7}$.
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