If `f(x)={((tan^-1(x+[x]))/([x]-2x)[x]ne0,,),(0[x]=0,,):}` where `[x]` denotes the greatest integer less than or equal to x, than `lim_(xto0) f(x)` is (a) `(-1)/2` (b) 1 (c) `(pi)/4` (d) Does not exist

Options

A. `-(1)/(2)`
B. `1`
C. `(pi)/(4)`
D. non-existent

Answer

D

Explanation

We have,
` lim_(xto0^-)f(x)=lim_(hto0)f(0-h)=lim_(hto0)lim_(hto0)f(-h)`
` rArr lim_(xto0^-)f(x)=lim_(xto0) (tan^-1(-h+[-h]))/([-h]+2h)`
` lim lim_(hto0) (tan^-1(-1-h))/(-1+2h)`
`rArr lim_(xto0^-)f(x)=lim_(hto0) (tan^-1(1+h))/(1-2h)=tan^-1=(pi)/(4)`
and ,
` lim_(xto0^+)f(x)=lim_(hto0) f(0+h)=lim_(hto0) f(h)`
` rArr lim_(xto0^+)f(x)=lim_(hto0)(tan^-1 (h+[h]))/([h]-2h)`
` rArr lim_(xto^+)f(x)=lim_(hto0) (tan^-1h)/(-2h)=-(1)/(2)[because [h]=0]`
Hence , `lim_(xto0) f(x)` does not exist.