Q:

What is the limit of this function as x goes to minus infinity?

I would like to compute the limit of:
$$\lim_{x\to -\infty} f(x)$$
where
$$ {f(x)=e^{-2x}(\cos(x)-\sin(x))}$$
My thought process was that since cosx and sinx are bounded between 1 and -1 and the exponential function goes to infinity the function will not converge. Is that correct and how can I prove it? 
Thanks!

A:

If diverge is the logical negation of converge, then you are right. But if diverge means divergent to infinity, then I am sorry but you are wrong. The limit does not exist at all, as WoframAlpha clearly shows.
For example, can you find a sequence $x_n \to -\infty$ such that $\cos x_n=\sin x_n$? Along this sequence, $f(x_n)=0$.
But you can easily find sequences $\{x_n\}$ such that $f(x_n) \to \pm\infty$.

