Question

Let \([x]\) denotes the greatest integer less than or equal to x and \(f(x) = [tan^2x]\), then which of the following is true ?

• A. \(\underset{x\rightarrow 0}{lim}f(x)\) does not exist
• B. \(f(x)\) is continuous at \(x = 0\)
• C. \(f(x)\) is not differentiable at \(x = 0\)
• D. \(f^'(0) = 1\)

Hint:

Near 0, tan squared x lies between 0 and 1, so the floor is 0.

Answer 1

The Correct Option is B

Step 1: Compute values near 0:
Take $x = 0.1$: $\tan^2 0.1 \approx 0.0101$, floor is 0. Take $x = -0.5$: $\tan^2 0.5 \approx 0.298$, floor 0.

Step 2: Limit:
The left-hand limit and right-hand limit are both 0, and $f(0) = 0$.

Step 3: Pick the true option:
That makes $f$ continuous at 0. The derivative is the limit of $0/h$, which is 0, so the options about non-existence of a limit, non-differentiability and $f'(0) = 1$ are all false.

Final Answer:
Option (B) is true. \[ \boxed{\text{(B)}} \]