Question

Evaluate the following limits,
where [.] represents greatest integer function and {.} represents fractional part function
(i) $\underset{x\to \frac{\pi }{2}}{lim}\left[sinx\right]$
(ii) $\underset{x\to 2}{lim}\left\{\frac{x}{2}\right\}$
(iii) $\underset{x\to \pi }{lim}sng\left[tanx\right]$
(iv) $\underset{x\to 1}{lim}si{n}^{-1}\left(lnx\right)$

Answer 1

$\left(1.\right)LHL={lim}_{h\to 0}\left(f\right(\frac{\pi }{2})-h)={lim}_{h\to 0}\left(sin\right[\left(\frac{\pi }{2}\right)-h\left)\right]={lim}_{h\to 0}\left[cosh\right]=0RHL={lim}_{h\to 0}\left(f\right(\frac{\pi }{2})+h)={lim}_{h\to 0}\left[sin\right(\frac{\pi }{2})+h)]={lim}_{h\to 0}[cosh]=0\therefore LHL=RHL=0$