Is $e^{2n}$ the dominant power in $\lim_{n\to\infty}\frac{e^{2n}+5}{3e^{2n}+10n}$? I am doing a root test and am not sure how to proceed. I want to say that it converges to $1/3$ based on the coefficients for the $e^{2n}$ terms in the numerator and denominator, since using L'Hospital's rule would just give increasingly complex limits, but I am not sure if this is valid.
Answer
Yes, in this case. Exponential functions grow faster than any polynomial or constant in the infinite limit, so you would be correct to suggest the limit is $1/3$.
You can see why the exponential function grows faster from the power series:
$$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$
So if there's a polynomial of degree $n$, $e^x$ has an expansion with terms of degree $n+1,n+2,n+3,$ etc., allowing it to easily dwarf the polynomial.
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