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  •  The integer  for which the <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{(\cos x-1)

 The integer \mathrm{n}  for which the \mathrm{\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}}   is a finite non-zero number is

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

Let \mathrm{l=\lim _{n \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}=\lim _{n \rightarrow 0} \frac{-2 \sin ^2 \frac{x}{2}\left(\cos x-e^x\right)}{x^n}}

For \mathrm{n=1, n=2,1=0}
For \mathrm{n=3},

\mathrm{l =\lim _{n \rightarrow 0}-2\left(\frac{\sin (x / 2)}{x / 2}\right)^2 \cdot \frac{\cos x-e^x}{x^2 \cdot x}=-\frac{1}{2} \lim _{n \rightarrow 0} \frac{\cos x-e^x}{x}}

\mathrm{=\frac{1}{2} \lim _{n \rightarrow 0}\left(-\sin x-e^x\right)=-\frac{1}{2}\left(-e^0\right)=\frac{1}{2}}

\mathrm{\therefore l}  is finite and non-zero. Hence \mathrm{n=3},

Posted by

Anam Khan

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