Question

Given $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^2}=2$, where [.] denotes the greatest integer function, then

• A. $\underset{x\to 0}{lim}\left[f\right(x\left)\right]=0$
• B. $\underset{x\to 0}{lim}\left[f\right(x\left)\right]=1$
• C. $\underset{x\to 0}{lim}\left[\frac{f\left(x\right)}{x}\right]$ does not exist
• D. $\underset{x\to 0}{lim}\left[\frac{f\left(x\right)}{x}\right]$ exists

Answer 1

Answer: (A), (C)

$\underset{x\to 0}{lim}\left[f\right(x\left)\right]=0$
$\underset{x\to 0}{lim}\left[\frac{f\left(x\right)}{x}\right]$ does not exist

Since $x^2$ > 0 and limit equals 2, f(x) must be a positive quantity.

Also, since $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^2}=2$, denominator $\to$ zero and limit is finite.

Therefore, f(x) must be approaching 0 or ${lim}_{x\to 0}f\left(x\right)=0^{+}$.
Hence, $\underset{x\to 0}{lim}\left[f\right(x\left)\right]=0$.

$\underset{x\to 0^{+}}{lim}\left[\frac{f\left(x\right)}{x}\right]=\underset{x\to 0^{+}}{lim}\left[x\frac{f\left(x\right)}{x^2}\right]=0$

and $\underset{x\to 0^{-}}{lim}\left[\frac{f\left(x\right)}{x}\right]$

$=\underset{x\to 0^{-}}{lim}\left[x\frac{f\left(x\right)}{x^2}\right]=-1$

Hence, $\underset{x\to 0}{lim}\left[\frac{f\left(x\right)}{x}\right]$ does not exist.