Let $a,b,c$ are non zero constant number then $\underset{r\to \infty }{\lim }\left[\frac{\cos \left({\displaystyle \frac{a}{r}}\right)-\cos \left({\displaystyle \frac{b}{r}}\right)\cos \left({\displaystyle \frac{c}{r}}\right)}{\sin \left({\displaystyle \frac{b}{r}}\right)\sin \left({\displaystyle \frac{c}{r}}\right)}\right]$ equals.

Given $l_1=\underset{x\to \frac{\pi }{4}}{\lim }{\cos }^{-1}\left[\sec \left(x-\frac{\pi }{4}\right)\right]; l_2=\underset{x\to \frac{\pi }{4}}{\lim }{\sin }^{-1}\left[\mathrm{cosec}\left(x+\frac{\pi }{4}\right)\right]$;

$l_3=\underset{x\to \frac{\pi }{4}}{\lim }{\tan }^{-1}\left[\cot \left(x+\frac{\pi }{4}\right)\right]; l_4=\underset{x\to \frac{\pi }{4}}{\lim }{\cot }^{-1}\left[\tan \left(x-\frac{\pi }{4}\right)\right]$

where $\left[x\right]$ denotes greatest integer function then which of the following limits exist?

where $\left[x\right]$ denotes step up function & $\left\{x\right\}$ fractional part function.