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Question Number 123398 by bemath last updated on 25/Nov/20

$$\underset{x\to 0}{\lim }\frac{\sqrt[3]{1+3x}-\sqrt{1+2x}}{x^2}?$$

Answered by Dwaipayan Shikari last updated on 25/Nov/20

$$\underset{x\to 0}{\lim }\frac{1+\frac{3x}{3}+\frac{9x^2}{2!}.(-\frac{2}{9})-1-\frac{2x}{2}+\frac{4x^2}{2!}.(\frac{1}{2}.\frac{1}{2})}{x^2}$$$$=\frac{-x^2+\frac{x^2}{2}}{x^2}=-\frac{1}{2}$$

Answered by TANMAY PANACEA last updated on 25/Nov/20

$${(1+x)}^n=1+nx+\frac{n(n-1)}{2!}{x}^2+o(thersterms$$$${(1+3x)}^{\frac{1}{3}}=1+\frac{1}{3}\left(3x\right)+\frac{\frac{1}{3}(\frac{1}{3}-1)}{2!}{\left(3x\right)}^2$$$$=1+x+\frac{\frac{-2}{9}}{2}\times 9x^2=1+x-x^2$$$${(1+2x)}^{\frac{1}{2}}=1+\frac{1}{2}\left(2x\right)+\frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}{\left(2x\right)}^2$$$$=1+x-\frac{1}{8}\times 4x^2$$$$N_r=(1+x-x^2)-(1+x-\frac{x^2}{2})=\frac{-1}{2}{x}^2$$$$\underset{x\to 0}{\lim }\frac{-x^2}{2x^2}=\frac{-1}{2}$$$$plschk$$

Answered by TANMAY PANACEA last updated on 25/Nov/20

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

Answered by liberty last updated on 25/Nov/20

$$\underset{x\to 0}{\lim }\frac{\sqrt[3]{1+3x}-\sqrt{1+2x}}{x^2}=$$$$\underset{x\to 0}{\lim }\frac{(1+\frac{\left(3x\right)}{3}+\frac{\left(1/3\right)(-2/3)}{2!}{\left(3x\right)}^2+o\left(x^2\right))-(1+\frac{\left(2x\right)}{2}+\frac{\left(1/2\right)(-1/2)}{2!}{\left(2x\right)}^2+o\left(x^2\right))}{x^2}=$$$$\underset{x\to 0}{\lim }\left(\frac{1+x-x^2-1-x+\frac{1}{2}{x}^2+o\left(x^2\right)}{x^2}\right)=$$$$\underset{x\to 0}{\lim }\left(\frac{x^2}{-2x^2}\right)=-\frac{1}{2}$$

Answered by Bird last updated on 25/Nov/20

$$letf\left(x\right)=\frac{{(1+3x)}^{\frac{1}{3}}-{(1+2x)}^{\frac{1}{2}}}{x^2}$$$$wehave{(1+3x)}^{\frac{1}{3}}=$$$$1+\frac{1}{3}\left(3x\right)+\frac{1}{3}(\frac{1}{3}-1)\times \frac{1}{2}{\left(3x\right)}^2+o(x^{3)}$$$$=1+x+\frac{1}{3}(-\frac{2}{3}).\frac{1}{2}\left(9x^2\right)+o\left(x^3\right)$$$$=1+x-x^2+o\left(x^3\right)$$$${(1+2x)}^{\frac{1}{2}}=1+\frac{1}{2}\left(2x\right)$$$$+\frac{1}{2}.\frac{1}{2}(\frac{1}{2}-1){\left(2x\right)}^2+o\left(x^3\right)$$$$=1+x-\frac{x^2}{2}+o\left(x^3\right)\Rightarrow$$$$f\left(x\right)\sim \frac{1+x-x^2-1-x+\frac{x^2}{2}}{x^2}\Rightarrow$$$$f\left(x\right)\sim -\frac{1}{2}\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=-\frac{1}{2}$$

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