Question

If `f(x) ={([x]^2+sin[x])/underset(0)([x])underset("for"[x]=0) "for" [x]ne 0` where `[x]` denotes the greatest integer function, then , `lim_(xto0)f(x)`, is
A. 1
B. 0
C. -1
D. non-existent

Answer 1

Correct Answer - D
We have ,
`lim_(xto0^-)[x]=-1and lim_(xto0^+)[x]=0`
`therefore lim_(xto0^-)f(x)=lim_(xto0^-)((-1)^2+sin(-1))/((-1))=-1+sin1`
and, `lim_(xto0^+)f(x)=lim_(xto0^+)[because f(x)=0 "for"0lex lt 1]`
So, `lim_(x to 0)f(x)` does not exist.