MHT CET · Maths · Limits
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
- A \(\lim _{x \rightarrow 4^{-}} f(x)\) exists, but \(\lim _{x \rightarrow 4^{+}} f(x)\) does not exist.
- B \(f(x)\) is continuous at \(x=4\).
- C \(\lim _{x \rightarrow 4^{+}} f(x)\) exists, but \(\lim _{x \rightarrow 4^{-}} f(x)\) does not exist.
- D Both \(\lim _{x \rightarrow 4^{-}} f(x)\) and \(\lim _{x \rightarrow 4^{+}} f(x)\) exist, but are not equal.
Answer & Solution
Correct Answer
(B) \(f(x)\) is continuous at \(x=4\).
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \lim _{x \rightarrow 4^{-}} f(x)=\left[4^{-}\right]-\left[\frac{4^{-}}{4}\right]=3-0=3 \text { and } \\ & \lim _{x \rightarrow 4^{+}} f(x)=\left[4^{+}\right]-\left[\frac{4^{+}}{4}\right]=4-1=3 \\ & f(4)=[4]-\left[\frac{4}{4}\right]=4-1=3 \\ & \because \lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)=f(4) \\ & \Rightarrow f(x) ; \text { continuous at } x=4\end{aligned}\)
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