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

The integer \small n for which  \small \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

\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}   writing expansion of all terms. 
 

\lim _{x \rightarrow 0} \frac{\left(1-\frac{x^2}{2 !}+\frac{x^4}{4 !}+\ldots-1\right)\left[1-\frac{x^2}{2 !}+\ldots-\left(1+x+\frac{x^2}{2} \ldots\right)\right]}{x^n}

=\lim _{x \rightarrow 0} \frac{\frac{x^3}{3 !}\left(1-\frac{x^2}{4 !}\right)\left(1+\frac{x^3}{3 !}+\frac{x^5}{5 !}\right)^{x^n}}{x^n}

When in reduces to   \lim 1 / 2 !(1-0+0)(1+0+0)=1 / 2
 

 

Posted by

jitender.kumar

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