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Question Number 85625 by jagoll last updated on 23/Mar/20

$$\underset{x\to 0}{\lim }\frac{{\mathrm{x}}^{\mathrm{n}}-{\left(\sin \mathrm{x}\right)}^{\mathrm{n}}}{{\left(\sin \mathrm{x}\right)}^{\mathrm{n}+2}}$$

Answered by john santu last updated on 23/Mar/20

$$byMaclaurinseries$$$$\sin x=x-\frac{x^3}{3!}+...$$$$\sin {}^{n}x=x^n{(1-\frac{x^2}{3!}+...)}^n$$$$letL=\underset{x\to 0}{\lim }\frac{x^n-x^n{(1-\frac{x^2}{6}+...)}^n}{x^{n+2}{(1-\frac{x^2}{6}+...)}^{n+2}}$$$$L=\underset{x\to 0}{\lim }\frac{\frac{{nx}^{n+2}}{6}-o\left(x^{n+4}\right)}{x^{n+2}-o\left(x^{n+6}\right)}$$$$L=\frac{n}{6}$$

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