This one really crushed my intuition. Let say a function $f$ grows faster than a function $g$ if $ \lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty $
Which of the following functions grows the fastest :
$2^{n/2}$
$3^{n/3}$
$5^{n/5}$
$1000^{1000/n}$
$10000^{10000/n}$
My bet would be on either function 1. or 5. but as it turns out, function 2. is growing the fastest.
After doing some calculations I was able to assure myself that that is really so. But my main doubt is still there. Why is this obvious? How can one intuitively explain this behavior?