The integer n for which \lim_{x \to 0} {(\cos x-1)(\cos x-e^{x})}/{x^{n}} is a finite non-zero number is

NIMCET 2008 Question Paper with solutions

Question : 37 of 120

Marks: +1, -0

The integer

n

for which

\lim ↙{x → 0} \;{(\cos x-1)(\cos x-e^{x})}/{x^{n}}

is a finite non-zero number is

Solution:

\lim ↙{x → 0}=\;{(\cos x-1)(\cos x-e^{x})}/{x^{n}}

=\lim ↙{x → 0} \;{(1-{x^{2}}/{2 !}+. . .-1)(1-{x^{2}}/{2 !}+. . .-(1+{x}/{1 !}+. . .))}/{x^{n}}

=\lim ↙{x → 0} \;{(-{x^{2}}/{2 !}+{x^{4}}/{4 !}+. . .)(-{x}/{1 !}-{2 x^{2}}/{2 !}-x^{3}/{3!}+{x^{4}}/{4 !} . . .)}/{x^{n}}

=\lim ↙{x → 0} \;{x^{3}(-{1}/{2 !}+{x^{2}}/{4 !}+. . .)(-1-{2 x}/{2 !}-{x^{2}}/{3 !} . . .)}/{x^{n}

which will be finite non-zero value, if

n=3

and the value is

\;{1}/{2}

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