Why is $\lim\limits_{x\to-\infty}\log [1 + \exp(-x)]+x=0$? 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].

See: http://www.wolframalpha.com/input/?i=Limit%5BLog%5B1%2BExp%5B-x%5D%5D%20%2B%20x%2C%20x%20-%3E%20Infinity%5D&amp;t=crmtb01

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.