Is $e^{2n}$ the dominant power in this limit?

Question

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 1

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.