Number of integral values of k for which
`lim_(xto1) sin^(-1)((k)/(log_(e)x)-(k)/(x-1))` exists is _________.

Options

A. `a=1,b=4`
B. `a=1b=-4`
C. `a=2,b=-3`
D. `a=2,b=3`

Answer

`(5)`

Explanation

We have `underset(xto1)limsin^(-1)((k)/(log_(e)x)-(k)/(x-1))`
Now, `underset(xto1)lim((k)/(log_(e)x)-(k)/(x-1))=kunderset(hto0)lim(h-log_(e)(1+h))/(h^(2))`
`=kunderset(hto0)lim(h-(h-(h^(2))/(2)+...))/(h^(2))=(k)/(2)`
Now `"sin"^(-1)(k)/(2)` is defined if`-1le(k)/(2)le1`
`implies" "-2lekle2`