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Q.

The integer n for which $\lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}}$ is a finite non-zero number is

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a

2

b

1

c

3

d

4

answer is C.

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Detailed Solution

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$\lim_{x \to 0} \frac{- (1 - \cos x) (\cos x - e^{x})}{x^{2} \cdot x^{n - 2}} = \frac{- 1}{2} \lim_{x \to 0} \frac{\cos x - e^{x}}{x^{n - 2}}$ $= \frac{- 1}{2} \cdot \lim_{x \to 0} \frac{- \sin x - e^{x}}{(n - 2) x^{n - 3}} ∴ n - 3 = 0 \Rightarrow n = 3$

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The integer n for which limx→0 (cos x-1)cos x-exxn is a finite non-zero number is