Find the value of _{h arrow 0} (a+h)² (a+h)-a² a/h

Correct Answer is : a2cos⁡a+2asin⁡aa² a+2 a aa2cosa+2asina

Correct Answer is : a2cos⁡a+2asin⁡aa² a+2 a aa2cosa+2asina

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COMEDK 2025 Morning Solved Paper

Question : 123 of 180

Marks: +1, -0

\lim\limits_{h \rightarrow 0} \frac{(a+h)^2 \sin (a+h)-a^2 \sin a}{h}

Solution:

Concept:

The given limit is recognized as the derivative of

f(x) = x^2 \sin x

x = a

Explanation:

We have

\lim\limits_{h\to 0} \frac{(a+h)^2 \sin(a+h) - a^2 \sin a}{h}

By definition of derivative, this equals

f'(a)

where

f(x)=x^2\sin x

Differentiate

f(x)

using the product rule:

f'(x) = \frac{d}{dx}(x^2) \cdot \sin x + x^2 \cdot \frac{d}{dx}(\sin x)

This gives

f'(x) = 2x \sin x + x^2 \cos x

Now evaluate at

x = a

f'(a) = 2a \sin a + a^2 \cos a

Thus, the limit simplifies to

a^2 \cos a + 2a \sin a

Answer:

The correct option is D:

a^2 \cos a + 2a \sin a

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