A) Both \[\underset{x\to 4-}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to 4+}{\mathop{\lim }}\,f(x)\] exist but are not equal
B) \[\underset{x\to 4-}{\mathop{\lim }}\,f(x)\] exists but \[\underset{x\to 4+}{\mathop{\lim }}\,f(x)\] does not exist
C) \[\underset{x\to 4+}{\mathop{\lim }}\,f(x)\] exists but \[\underset{x\to 4-}{\mathop{\lim }}\,f(x)\]does not exist
D) f is continuous at x = 4
Correct Answer: D
Solution :
\[f(x)=[x]-\left[ \frac{x}{4} \right]\] \[\underset{x\to 4+}{\mathop{\lim }}\,f(x)=\underset{x\to 4+}{\mathop{\lim }}\,\left( \left( [x]-\left[ \frac{x}{4} \right] \right) \right)=4-1=3\] \[\underset{x\to 4-}{\mathop{\lim }}\,f(x)=\underset{x\to 4-}{\mathop{\lim }}\,\left( [x]-\frac{x}{4} \right)=3-0=3\]\[f(x)=3\] \[\therefore \] continuous at \[x=4\]
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