I was going through a question today.
$$\lim_{x\to \infty} x^5 \left\lfloor \frac{1}{x^3}\right\rfloor $$ where $\lfloor \cdot \rfloor$ represents the G.I.F. (floor function).
Without actually trying to solve the question, I tried to use some manual approximations. Like I assumed as $x$ is tending to infinity, It will be a considerably large number but finite. So undoubtedly $$ \frac{1}{x^3} \ll 1 $$
therefore,
$$ \left\lfloor \frac{1}{x^3} \right\rfloor =0 $$
So what we are actually doing is multiplying a very large number but finite $x^5$ to a definite zero. So I concluded that this quantity will tend to zero as $x$ tends to infinity. But the answer is not zero.
What am I missing out here? Thanks.