ISRO 2009- ECE Calculus

Question

f(x) = [$ tan ^2$ x]  ( [ ] stands for greatest integer function)

a) f(x) continuous at x = 0

b) limit f(x) does not exist as x tend to 0

c) f '(0) = 1

d) f(x) not derivable at x = 0

Answer 1

In the neighbourhood of 0 the value of tan^2x lies between 0 to 1. So the GIF will be a constant value of 0

(Consider -pi/4 to pi/4 as the neighbourhood of 0)    -pi/4 < x < pi/4   => -1 < tanx < 1 => 0 < tan^2 x < 1  => [tan^2 x] = 0

So it value is constant ie = => Continous => Limit Exists .
constant function  is differentiable => f’(0)=0

so A is true , B is false as limit exists and equal to 0 , C is false as f’(0)=0, D is also false