If {x} denotes the fractional part of x, then `underset(x to 0)(lim) ({x})/(tan {x})` is equal to

Options

A. 1
B. 0
C. -1
D. none of these

Answer

D

Explanation

We have
` lim_(xto0^-)({x})/(tan{x})`
` =lim_(xto0^-)(x-[x])/(tan (x-[x]))=lim_(xto0^-)(x+1)/(tan(x+1))=(1)/(tanx+1)=(1)/(tan1)=cot 1`
and
` lim_(xto0^-)({x})/(tan{x})=lim_(xto0^+)(x-[x])/(tan(x-[x]))=lim_(xto0^+)(x)/(tanx)=1`
Clearly , `lim_(xto0^-)({x})/(tan{x}) ne lim_(xt0^+)({x})/(tan{x})`
So , `lim_(xto0) ({x})/(tan{x})` does not exist.