Question 2

Let f and g be real valued functions defined on interval (–1, 1) such that g(x) is continuous, g(0) ≠ 0, g'(0) = 0, g(0) ≠ 0, and f(x) = g(x) sin x.
Statement-1 : lim(x-->0) [g(x) cot x – g(0) cosec x] = f (0) and
Statement-2 : f '(0) = g(0)

Options

1. Statement-1 is True, Statement-2 is True; statement-2 is a correct explanation for statement-1

2. Statement-1 is True, Statement-2 is True; statement-2 is NOT a correct explanation for statement-1

3. Statement-1 is True, Statement-2 is False

4. Statement-1 is False, Statement-2 is True

To tackle this question, we need to analyze the given statements about the functions \( f \) and \( g \). We know that \( f(x) = g(x) \sin x \) and we have specific conditions on \( g \) at \( x = 0 \). Let's break down each statement step by step.

Examining Statement-1

Statement-1 claims that:

lim(x→0) [g(x) cot x – g(0) cosec x] = f ''(0)

To evaluate this limit, we first need to find \( f''(0) \). We can use the product rule to differentiate \( f(x) \):

• First derivative:

  Using the product rule, we have:

  f'(x) = g'(x) sin x + g(x) cos x

• Second derivative:

  Now, differentiating again:

  f''(x) = g''(x) sin x + 2g'(x) cos x - g(x) sin x

  Evaluating at \( x = 0 \):

  f''(0) = g''(0) sin(0) + 2g'(0) cos(0) - g(0) sin(0) = 0 + 0 - 0 = 0

Now, let's analyze the limit:

As \( x \to 0 \), we know that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \). Therefore, we can rewrite the expression:

g(x) cot x - g(0) cosec x = g(x) \frac{\cos x}{\sin x} - g(0) \frac{1}{\sin x}

Factoring out \( \frac{1}{\sin x} \):

lim(x→0) [g(x) cos x - g(0)] / sin x

Using L'Hôpital's Rule (since both the numerator and denominator approach 0 as \( x \to 0 \)), we differentiate the numerator and denominator:

Numerator: g'(x) cos x - g(x) sin x

Denominator: cos x

Evaluating this limit gives us \( f''(0) \), confirming that Statement-1 is indeed true.

Analyzing Statement-2

Statement-2 states that:

f '(0) = g(0)

We already found \( f'(0) \) earlier:

f'(0) = g'(0) sin(0) + g(0) cos(0) = 0 + g(0) = g(0)

This means Statement-2 is also true.

Connecting the Statements

Now, we need to determine if Statement-2 provides a correct explanation for Statement-1. While both statements are true, Statement-2 does not explain the limit in Statement-1, which is more complex and involves the second derivative. Therefore, we conclude that:

Final Evaluation

Both statements are true, but Statement-2 does not serve as a correct explanation for Statement-1. The correct answer is:

Option 2: Statement-1 is True, Statement-2 is True; statement-2 is NOT a correct explanation for statement-1.