lim-x-0-x-n-sin-n-x-x-n-sin-n-x-is-non-zero-finite-then-find-n-

Question Number 141965 by gsk2684 last updated on 25/May/21

\lim_{x \to 0} \frac{x^{n} \sin^{n} x}{x^{n} - \sin^{n} x} i s n o n - z e r o f i n i t e,

t h e n f i n d n ?

Commented by gsk2684 last updated on 25/May/21

s o l u t i o n p l e a s e

Answered by Dwaipayan Shikari last updated on 25/May/21

\lim_{x \to 0} s i n x \sim (x - \frac{x^{3}}{3!})

\lim_{x \to 0} \frac{x^{n} x^{n}}{x^{n} - {(x - \frac{x^{3}}{6})}^{n}} = \frac{x^{n}}{1 - {(1 - \frac{x^{2}}{6})}^{n}} = \frac{x^{n}}{\frac{{n x}^{2}}{6} + O (x^{4})} = \frac{6}{n} . \frac{x^{n}}{x^{2}}

O (x^{4}) \to 0

t o g e t a n o n z e r o f i n i t e v a l u e x^{n} = x^{2} t h e n n = 2

Commented by gsk2684 last updated on 25/May/21

t h a n k y o u

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