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Question Number 172361 by Mikenice last updated on 25/Jun/22

Answered by mr W last updated on 26/Jun/22

$$=\underset{x\to 0}{\lim }\frac{x-\frac{x^3}{3}+o\left(x^5\right)-(x+\frac{x^3}{6}+o\left(x^5\right))}{x^3}$$$$=\underset{x\to 0}{\lim }\frac{-(\frac{1}{3}+\frac{1}{6})x^3+o\left(x^5\right)}{x^3}$$$$=-(\frac{1}{3}+\frac{1}{6})$$$$=-\frac{1}{2}$$

Commented by bagjagugum123 last updated on 26/Jun/22

$$WhatisoSir?$$

Commented by mr W last updated on 26/Jun/22

$$o\left(x^5\right)meansfollowingtermsareof$$$$5^{th}orderorhigher.$$

Commented by mr W last updated on 26/Jun/22

Commented by bagjagugum123 last updated on 26/Jun/22

$$ThankyouSir$$

Answered by mahdipoor last updated on 25/Jun/22

$$=\underset{x\to 0}{\lim }\frac{\frac{1}{x^2+1}-\frac{1}{\sqrt{1-x^2}}}{3x^2}=\frac{\sqrt{1-x^2}-(1+x^2)}{3x^2(x^2+1)\sqrt{1-x^2}}=$$$$=\frac{(1-x^2)-{(1+x^2)}^2}{3x^2(x^2+1)\sqrt{1-x^2}(\sqrt{1-x^2}+(1+x^2))}=$$$$=\frac{-x^2(x^2+3)}{3x^2(x^2+1)\sqrt{1-x^2}(\sqrt{1-x^2}+(1+x^2))}=$$$$=-\frac{1}{2}$$$$$$

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