Question

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

• A. $\underset{\text{x $\rightarrow$ 0}}{\ce{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 (x) = 1

Answer 1

Answer: (B)

Check the continuity of the function
f (x) = [tan$^2$ x] at x = 0.
L.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^-$}}{\ce{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{\ce{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{\ce{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
R.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^+$}}{\ce{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{\ce{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{\ce{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
Now, determine the value of f(x) at x = 0.
f (0) = [tan$^2$ 0] = [0] = 0
Hence, f (x) is continuous at x = 0.