Limit using the monotone convergence theorem

Show using the monotonic convergence theorem that the limit below convergence, in other words, analytically demonstrate that the limit below convergence:

$$ lim_ {x to infty} frac {5 ^ x} {2 ^ {n ^ 2}} $$

I tried to do: $$ lim_ {x to infty} frac {x cdot ln {5}} {x ^ 2 cdot ln {2}} = 0 $$ but I do not know if this is allowed by the theorem.

analysis: if $ f (x) $ is monotone and $ int_a ^ {+ infty} f (x) mathrm {d} x $ is convergent, then $ lim_ {x rightarrow + infty} xf (x ) = 0 $

Yes $ f (x) $ it is monotonous and $ int_a ^ {+ infty} f (x) mathrm {d} x $ is convergent then test
$$ lim_ limits {x rightarrow + infty} xf (x) = 0 $$

I think of the second theorem of the mean value for integrals
$$ int_X ^ {X & # 39;} f ​​(x) mathrm {d} x = f (X) int_X ^ xi f (x) + f (X & # 39;) int_ xi ^ { X & # 39;} $$
because of the monotony of $ f (x) $. Y
$$ forall epsilon> 0, | int_X ^ {X & # 39;} f ​​(x) mathrm {d} x | = | f (X) int_X ^ { xi} mathrm {d} x + f (X & # 39;) int_ xi ^ {X & # 39;} mathrm {d} x | = | (X & # 39; – xi) f (X & # 39;) + ( xi – X) f (X) | $$
Y
$$ | X & # 39; f (X & # 39;) – Xf (X) | = | (X & # 39; – xi) f (X & # 39;) + ( xi – X) f (X) + xi f (X & # 39;) – xi f (X) | $$

But I have no idea to continue.

Algorithms – Find the direction of the monotone polygon (if it exists) in linear time

Implement a program that verifies if there is an address where a simple polygon is monotonous and, in that case, notify that address.
Upper limits: O (n) time, where n is the size of
the polygon

The vertices are in counter-clockwise order.
I have no idea how to solve this problem, if you do not know the solution, even the idea could help you have something to work on.

I have tried to classify the vertices in (start, end, division, combination, regular) as in triangulation by some random direction and do something with it. Because if there are no divisions or mergers, the cop is monotonous in that direction. But I think I came to an impasse.