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The integer \mathrm{n} for which \mathrm{\lim _{x \rightarrow 0}\left\{\frac{(\cos x-1)\left(\cos x-e^2\right)}{x^n}\right\}} is a finite, non-zero real number, is

 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

We know that

\mathrm{ \cos x=1+\frac{x^2}{2}+\frac{x^4}{4}+\ldots \text { ad inf. } }............(1)

\mathrm{ e^x=1+x+\frac{x^2}{2}+\frac{x^3}{\underline{3}}+\cdots \text { ad inf. } }.............(2)

\mathrm{ \left.\therefore(\cos x-1)=x^2 \mid \frac{1}{12}+\frac{x^2}{44}+\ldots\right] \text { and } }

\mathrm{ \left(\cos x-e^2\right)=-x\left[1+\frac{x^2}{2}+\ldots .\right] \text {. } }

\mathrm{ \left.\therefore \lim _{x \rightarrow 0} \mid \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}\right] }

\mathrm{=-\lim _{x \rightarrow 0}\left\{\frac{\left(\frac{1}{2}+\frac{x^2}{4}+\ldots\right)\left(1+\frac{x^2}{2}+\ldots\right)}{x^{n-3}}\right\} }

\mathrm{\Rightarrow n=3 \text {. } }

Hence option 3 is correct.










 

Posted by

Devendra Khairwa

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