JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If \(f\left( x \right) = \left[ x \right] - \left[ {\frac{x}{4}} \right],\,x \in R\) , where \([x]\) denotes the greatest integer function, then
- A Both \(\mathop {\lim }\limits_{x \to 4 - } \,f\left( x \right)\) and \(\mathop {\lim }\limits_{x \to 4 + } \,f\left( x \right)\) exist but are not equal
- B \(\mathop {\lim }\limits_{x \to 4 - } \,f\left( x \right)\) exists but \(\mathop {\lim }\limits_{x \to 4 + } \,f\left( x \right)\) does not exist
- C \(\mathop {\lim }\limits_{x \to 4 + } \,f\left( x \right)\) exists but \(\mathop {\lim }\limits_{x \to 4 - } \,f\left( x \right)\) does not exist
- D \(f\) is continuous at \(x = 4\)
Answer & Solution
Correct Answer
(D) \(f\) is continuous at \(x = 4\)
Step-by-step Solution
Detailed explanation
\(f\left( x \right) = \left[ x \right] - \left[ {\frac{x}{4}} \right]\) \(\mathop {\lim }\limits_{x \to {4^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {4^ + }} \left( {\left[ x \right] - \left[ {\frac{x}{4}} \right]} \right) = 4 - 1 = 3\)…
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