Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$ Show that
$$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$
How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm.

If $x \to +\infty$, By using L'Hospital's rule:
$$=\lim_{x\to +\infty}\frac{e^x}{1+e^x}=1-\lim_{x\to +\infty}\frac{1}{1+e^x}=1$$
If $x\to -\infty$, It equals 0
So I'm sorry to tell you that this limit does not exist!
