Okay, so I read this somewhere that,
$$ \lim_{x \to 0^+} \left[ \frac{\sin x}{x} \right] = 0 $$
Where, [] denotes the greatest integer function.
But, on the other hand, this is also true,
$$ [0.9999…] = 1 $$
Aren't these two contradictory? I mean if,
$$ \lim_{x \to 0^+} \frac{\sin x}{x} = 1 $$
and
$$ 0.9999… = 1 $$
Then why is the greatest Integer function behaving differently for these two functions?
Best Answer
$0.999\ldots$ isn't a sequence / function you take the limit of. It is a fixed number, and it's equal to $1$ in value (more strictly: If you allow it to represent a value, then any value except $1$ will get you into inconsistencies). Round it down all you like, that doesn't change. On the other hand, we do have $$ \lim_{n \to \infty}\left[0.\underbrace{999\ldots99}_{n\text{ times}}\right] = 0 $$ As for why $\lim_{x \to 0^+} \left[ \frac{\sin x}{x} \right] = 0$, that's simply because for all non-zero $x$, we have $\left[ \frac{\sin x}{x} \right] = 0$, so of course the limit is going to be $0$ as $x \to 0$.