Find $\lim_{x\to-\infty}{x+e^{-x}}$ I have this exercise in my worksheet:
$$\lim_{x\to-\infty}{x+e^{-x}}$$
I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
Negative numbers make me nervous, so let $t=-x$. We want
$$\lim_{t\to \infty} (e^t-t).$$ 
The answer is obvious, $e^t$ is much larger than $t$ if $t$ is large. If you want to be formal, after a (short) while $t\lt \frac{e^t}{2}$, so after a short while $e^t-t\gt \frac{1}{2}e^t$.
The given expression does not have a limit. That is to say, it does not converge on a value as $x$ grows large, negative. 

As $x$ grows large and negative, the $x$ term grows large and negative. The $e^{-x}$ term grows large and positive, and its growth is faster than that of the $x$ term, because it is an exponential function whereas $x$ is polynomial. Exponentials grow faster than polynomials of all orders. Therefore, the exponential term dominates and the overall expression grows large, and positive. 

However, it does so without limit.  We cannot say that the limit is $+\infty$, because that is nonsense. A limit, when it exists, is a number, whereas $+\infty$ is not a number and does not substitute for a limit that doesn't exist.