Question

Let f and g be real valued functions defined on (– 1, 1) such that g"(x) is continuous g(0) ≠ 0, g'(0) = 0, g"(0) ≠ 0 and f(x) = g(x)sinx

Assertion (A): lim_(x→0)  (g(x) cotx – g(0) cosecx) = f"(0) 

Reason(R): Because f'(0) = g(0)

Answer 1

We have f(x) = g(x)sinx

⇒ f'(x) = g(x)cos x + g'(x)sinx

⇒ f"(x) = g'(x)cos x – g(x)sinx + g'(x)cos x + g"(x)sinx

Now, f'(0) = g(0) + g'(0). 0 = g(0)

So Statement-II is true and f"(0) = 2g'(0)

Now, lim_(x→0)  (g(x)cot x – g(0)cosecx)

Thus, statement-I is also true.

But statement-II is not a correct explanation for statement-I