If $f(x)=[x]-\left[\frac{x}{4}\right], x \in R$ Where $[x]$ denotes the greatest integer less than or equal…

If $f(x)=[x]-\left[\frac{x}{4}\right], x \in R$ Where $[x]$ denotes the greatest integer less than or equal to $x$, then
  1. $\lim _{x \rightarrow 4^{-}} f(x)$ exists, but $\lim _{x \rightarrow 4^{+}} f(x)$ does not exist.
  2. $f(x)$ is continuous at $x=4$.
  3. $\lim _{x \rightarrow 4^{+}} f(x)$ exists, but $\lim _{x \rightarrow 4^{-}} f(x)$ does not exist.
  4. Both $\lim _{x \rightarrow 4^{-}} f(x)$ and $\lim _{x \rightarrow 4^{+}} f(x)$ exist, but are not equal.

Solution

$\begin{aligned} & \lim _{x \rightarrow 4^{-}} f(x)=\left[4^{-}\right]-\left[\frac{4^{-}}{4}\right]=3-0=3 \text { and } \\ & \lim _{x \rightarrow 4^{+}} f(x)=\left[4^{+}\right]-\left[\frac{4^{+}}{4}\right]=4-1=3 \\ & f(4)=[4]-\left[\frac{4}{4}\right]=4-1=3 \\ & \because \lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)=f(4) \\ & \Rightarrow f(x) ; \text { continuous at } x=4\end{aligned}$

Asked in: MHT CET 2022 (10 Aug Shift 2)

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