Is the limit $\displaystyle\lim_{x\rightarrow0}\frac{\sin{[x]}}{[x]}$ a one sided limit or not? Here $[\, \cdot\, ]$ is the greatest integer function.
According to me the right hand limit will be not defined and the left hand limit will be $\dfrac{-\sin1}{-1}$.
You're completely correct.
The function is undefined for $0<x<1$, so the limit from the right is meaningless.
On the other hand the function is continuous (and constant) in the interval $(-1,0)$, so the limit from the left exists and is $\sin1$.