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Given lim(x to 0)(f(x))/(x^(2))=2, where...

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

A

`(1)/(3)`

B

`(1)/(4)`

C

`(1)/(2)`

D

`(2)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
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.
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