if f(x) = [x]
Question:

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

 

  1. (1) $f$ is continuous at $x=4$.

  2. (2) $\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist.

  3. (3) Both $\lim _{x \rightarrow 4-} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal.

  4. (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 $\mathrm{x}=4$

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