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**Illustration 1: Let [.] denotes the greatest integer function and f(x) = [tan ^{2}x], then does the limit exist or is the function differentiable or continuous at 0? (1999)**

Now, -45Â°< x < 45Â°

tan(-45Â°)< tanx < tan45Â°

-tan 45Â°< tan x < tan 45Â°

-1< tan x <1

So, 0 <tan

[tan

So, f(x) is zero for all values of x form x = -45Â° to 45Â°.

Hence, f is continuous at x =0 and f is also differentiable at 0 and has a value zero.

**Illustration 2: A function is defined as follows:****f(x)= x ^{3} , x^{2}< 1**

= limhâ†’0 (f(1-h)-f(1))/h

= limhâ†’0 (1+h-1)/h = 1

and L.H.D. = Lf'(1)= lim

= lim

= lim

?Rf'(1)â‰ Lf'(1)â‡’ f(x) is not differentiable at x=1.

**Illustration 3:If y = (sin-1x) ^{2} + k sin-1x, show that (1-x^{2}) (d^{2 }y)/dx^{2} - x dy/dx = 2**

Differentiating both sides with respect to x, we have

dy/dx = 2(sin-1 x)/âˆš(1-x

â‡’(1-x

Differentiating this with respect to x, we get

(1-x

â‡’(1-x

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