Compute the limit of $ \lim_{n \uparrow \infty} [f(x)e^{nx} + g(x)]/(e^{xn}+1)$, with $ x>0$ and $g(x)$ bounded

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Compute the limit

$$\lim_{n\to\infty}\frac{f(x)e^{nx}+g(x)}{e^{nx}+1},$$

where $x>0$ and $g$ is bounded.

Got no clue on this. A book claims f(x) as its limit without any hints and the only I see that could happen is when f=g. But I guess that doesn't work!

Answer 1

Hint: Rewrite the expression in the limit as

$$\frac{f(x)e^{nx}}{e^{nx}+1}+\frac{g(x)}{e^{nx}+1}=\frac{f(x)}{1+\frac{1}{e^{nx}}}+\frac{g(x)}{e^{nx}+1}$$

Can you compute the limit of each term in the sum above?

Answer 2

Simply $g(x) $ is bounded finitely. Thus: $$\frac{f(x)+\frac{g(x)}{e^{nx}}}{1+\frac{1}{e^{nx}}}$$

as $n\to\infty$, $e^{nx}\to\infty$ Thus it leaves you with $f(x)$