What is $\displaystyle \lim_{t->\infty}\frac{x^{t}}{\int_{0}^{\infty}u^{t}e^{-u}du}$ ?
One "cheap" way to do this is to assume there is one specific answer, independent of x. If the answer does not depend on x, take x= 0 and immediately the sequence is 0, 0, 0, 0,...
If x is not 0, we can at least assume x> 0 since otherwise $\displaystyle x^t$ is not defined for some t. And it is well known that t! dominates any x to the t power. One way of looking at it is this: the numerator and denominator both include t factors. In the numerator, the factor is always x. In the denominator the factors are from 1 up to t so for large enough t, the denominator is larger than the numerator- and as t gets larger the difference gets greater.