Given `("lim")_(xvec0)(f(x))/(x^2)=2,w h e r e[dot]` denotes the greatest integer function, then (a) `("lim")_(xvec0)[f(x)]=0` (b) `("lim")_(xvec0)[f(x)]=1` (c)`("lim")_(xvec0)[(f(x))/x]` does not exist (d)`("lim")_(xvec0)[(f(x))/x]` exists

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(1)/(2)`
D. `(2)/(3)`

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.