Question

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

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

Answer 1

Answer: (D)

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