if f(x) = [x]

Download now India's Best Exam Prepration App Class 8-9-10, JEE & NEET
Video Lectures	Live Sessions
Study Material	Tests
Previous Year Paper	Revision

Video Lectures

Live Sessions

Study Material

Tests

Previous Year Paper

Revision

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

(1) $f$ is continuous at $x=4$.
(2) $\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist.
(3) Both $\lim _{x \rightarrow 4-} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal.
(4) $\lim _{x \rightarrow 4-} f(x)$ exists but $\lim _{x \rightarrow 4+} f(x)$ does not exist.

Correct Option: 1

Solution:

L.H.L. $\lim _{x \rightarrow 4^{-}}\left([x]-\left[\frac{x}{4}\right]\right)=3-0=3$

R.H.L. $\lim _{x \rightarrow 4^{+}}[x]-\left[\frac{x}{4}\right]=4-1=3$

$f(4)=[4]-\left[\frac{4}{4}\right]=4-1=3$

$\because \mathrm{LHL}=f(4)=\mathrm{RHL}$

$\therefore \mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=4$