The integer n for which lim x 0 ( cos x - 1 ) ( cos x - e x ) x n is a finite non-zero number is [2002]

Question

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 [2002]

Smart Achievers · Accepted Answer

(3)

$\lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}}$

$= \lim_{x \to 0} \frac{(1 - \cos x) (1 + \cos x) (e^{x} - \cos x)}{x^{n} (1 + \cos x)}$

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

$= 1^{2} \cdot \frac{1}{2} \lim_{x \to 0} \frac{e^{x} - \cos x}{x^{n - 2}}$

$= \frac{1}{2} \lim_{x \to 0} \frac{e^{x} + \sin x}{(n - 2) x^{n - 3}}$ (Using L'Hospital rule)

For this limit to be finite, $n - 3 = 0$

$∴$ $n = 3$

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