Question

Let \([\,]\) denote the greatest integer function and \(f(x)=[\tan^2 x]\). Then

• A. \(\lim_{x\to 0} f(x)\) does not exist
• B. \(f(x)\) is continuous at \(x=0\)
• C. \(f(x)\) is not differentiable at \(x=0\)
• D. \(f(x)=1\)

Hint:

If a function becomes constant in a neighbourhood around a point, it is continuous (and differentiable) there.

Answer 1

The Correct Option is B

Step 1: Evaluate \(f(0)\).
\[ f(0)=[\tan^2 0]=[0]=0 \]
Step 2: Check behaviour near \(x=0\).
For small \(x\):
\[ \tan x \approx x \Rightarrow \tan^2 x \approx x^2 \]
So for \(x\) close to 0:
\[ 0 \le \tan^2 x<1 \]
Hence:
\[ [\tan^2 x]=0 \]
Step 3: Compute limit.
\[ \lim_{x\to 0} f(x)=\lim_{x\to 0} [\tan^2 x]=0 \]
Step 4: Compare with \(f(0)\).
\[ \lim_{x\to 0} f(x)=0=f(0) \]
So \(f(x)\) is continuous at \(x=0\).
Final Answer:
\[ \boxed{\text{(B) } f(x)\text{ is continuous at }x=0} \]