<s>[INST] Compute $ \lim_{n \to \infty} \frac{5^n}{5^n + 3^n} $ How can I compute the next limit of sequences:
$$ \lim_{n \to \infty} \frac{5^n}{5^n + 3^n} $$
It is $\frac{\infty}{\infty}$ indeterminate when evaluating in $n$, but using L'Hopital never eliminates the $5^n, 3^n$ terms. But because the denominator is greater than the numerator, it seems that the limit should be 1 or zero.
 [/INST] $$\lim_{n \to \infty} \frac{5^n}{5^n + 3^n} = \lim_{n \to \infty} \frac{5^n}{5^n\left(1 + \left(\frac{3}{5}\right)^n\right)} = \lim_{n \to \infty} \frac{1}{1 + \left(\frac{3}{5}\right)^n} = 1,$$
since $\left(\frac{3}{5}\right)^n$ goes to $0$ as $n$ goes to $+\infty$.
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