Could someone explain why
$\lim\limits_{x\to-\infty}\log [1 + \exp(-x)]+x=0$
Could someone explain why
$\lim\limits_{x\to-\infty}\log [1 + \exp(-x)]+x=0$
Here is a one-line (full) explanation: $\log(1+\mathrm e^{-x})+x=\log((1+\mathrm e^{-x})\cdot\mathrm e^{x})=\log(\mathrm e^{x}+1)\underset{x\to-\infty}{\longrightarrow}\log(0+1)=0$
I think you may have meant the exponential function, which is Exp[-x].
Do you see why it is infinity, just by inspection?
The exponential approaches zero, Log[1] is zero and the remaining terms grows to infinity.
If this is acceptable, you should accept the answer.