Given `lim_(x to 0)(f(x))/(x^(2))=2`, where `[.]` denotes the greatest integer function, then which options are correct?

Options

A. (a) `lim_(x to 0)[f(x)]=0`
B. (b) `lim_(x to 0)[f(x)]=1`
C. (c) `lim_(x to 0)[f(x)/(x)]` do e s not exist
D. (d) `lim_(x to 0)[f(x)/(x)]` exist

Answer

A, C

Explanation

Since `x^(2)gt0` and limit equals `2,f(x)` must be a positive quantity. Also, since `underset(xto0)lim(f(x))/(x^(2))=2`. Denominator `to` zero and limit is finite. Therefore, `f(x)` must be approaching zero or `underset(xto0)lim[f(x)]=0^(+)`.
Hence, `underset(xto0)lim[(f(x))/(x)]=0^(+)`.
`underset(xto0^(+))lim[(f(x))/(x)]=underset(xto0^(+))lim[x(f(x))/(x^(2))]=0`
and `underset(xto0^(-))lim[(f(x))/(x)]=underset(xto0^(-))lim[x(f(x))/(x^(2))]=-1`
Hence, `underset(xto0)lim[(f(x))/(x)]` does not exist.