Question

If $f(x) = [x] - \left[ \frac{x}{4} \right] , x \in R$, where [x] denotes the greatest integer function, then :

• A. Both $\displaystyle\lim_{x \to 4-} f(x)$ and $\displaystyle\lim_{x \to 4+} f(x)$ exist but are not equal
• B. $\displaystyle\lim_{x\to 4-} f(x)$ exists but $\displaystyle\lim_{x \to 4+} f(x)$ does not exist
• C. $\displaystyle\lim_{x\to 4 +} f(x) $ exists but $\displaystyle\lim_{x\to 4-} f(x)$ does not exist
• D. $f$ is continuous at $x = 4$

Answer 1

Answer: (D)

$f\left(x\right) =\left[x\right] - \left[\frac{x}{4}\right] $
$ \lim_{x\to4+} f\left(x\right) =\lim_{x\to4+} \left(\left(\left[x\right] - \left[\frac{x}{4}\right]\right)\right) = 4-1 =3 $
$ \lim_{x\to4+} f\left(x\right)= \lim_{x\to4-} \left(\left[x\right]- \frac{x}{4}\right) =3-0 =3 $
$ f\left(x\right) = 3 $
$ \therefore $ continuous at x = 4