Question

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

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

Answer 1

Answer: (B)

$f (x)$ is continuous at $x = 0$

Answer 2

We have $f (x) = [\tan^2 x]$
tan x is an increasing function for $- \frac{\pi}{4} < x < \frac{\pi}{4}$
$\therefore \, \tan\left(- \frac{\pi}{4}\right) < \tan x < \tan\left(\frac{\pi}{4}\right)$
$\Rightarrow-1 < \tan x < 1$
$\Rightarrow 0 < \tan^{2} x < 1$
$\Rightarrow \left[\tan^{2} x\right] = 0$
Hence, $\displaystyle \lim_{x \to0} f\left(x\right) =\displaystyle \lim_{x \to0} \left[\tan^{2} x\right] = 0$
Also $f\left(0\right) = 0$
$\therefore f\left(x\right)$ is continuous at $x = 0$