The integer n for whichlimx→0 (cos⁡x−1)cos⁡x−exxnis a finite nonzero number is

L=limx→0 (cos⁡x−1)cos⁡x−exxn=limx→0 (1−cos⁡x)(1+cos⁡x)-cos⁡x+ex(1+cos⁡x)xn=limx→0 sin⁡xx21−cos⁡xx+ex−1xxn−311+cos⁡xL is finite nonzero. Then n=3( as for n=1,2,L=0, and for n=4,L=∞)

The integer n for whichlimx→0 (cos⁡x−1)cos⁡x−exxnis a finite nonzero number is

L=limx→0 (cos⁡x−1)cos⁡x−exxn=limx→0 (1−cos⁡x)(1+cos⁡x)-cos⁡x+ex(1+cos⁡x)xn=limx→0 sin⁡xx21−cos⁡xx+ex−1xxn−311+cos⁡xL is finite nonzero. Then n=3( as for n=1,2,L=0, and for n=4,L=∞)

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

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Detailed Solution

$\begin{array}{l} L = lim_{x \to 0} \frac{(\cos x - 1) (\cos x - e^{x})}{x^{n}} = lim_{x \to 0} \frac{(1 - \cos x) (1 + \cos x) (- \cos x + e^{x})}{(1 + \cos x) x^{n}} \\ = lim_{x \to 0} \frac{{(\frac{\sin x}{x})}^{2} (\frac{1 - \cos x}{x} + \frac{e^{x} - 1}{x})}{x^{n - 3}} \frac{1}{1 + \cos x} \end{array}$
L is finite nonzero. Then $n = 3 (as for n = 1, 2, L = 0, and for n = 4, L = \infty)$