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Question Number 83323 by john santu last updated on 01/Mar/20

$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{(3\mathrm{x}-2)(\mathrm{x}-\sqrt{2})}-\mathrm{x}\sqrt{3}-\sqrt{2}=?$$$$$$

Commented by abdomathmax last updated on 01/Mar/20

$$letf\left(x\right)=\sqrt{(3x-2)(x-\sqrt{2})}-x\sqrt{3}-\sqrt{2}$$$$\Rightarrow f\left(x\right)=\sqrt{3x^2-3\sqrt{2}x-2x+2\sqrt{2}}-x\sqrt{3}-\sqrt{2}$$$$=\sqrt{3x^2-(3\sqrt{2}+2)x+2\sqrt{2}}-x\sqrt{3}-\sqrt{2}$$$$at+\mathrm{\infty }$$$$f\left(x\right)=x\sqrt{3}\sqrt{1-\frac{3\sqrt{2}+2}{3x}+\frac{2\sqrt{2}}{3x^2}}-x\sqrt{3}-\sqrt{2}$$$$\sim x\sqrt{3}\{1+\frac{1}{2}(-\frac{3\sqrt{2}+2}{3x}+\frac{2\sqrt{2}}{3x^3})\}-x\sqrt{3}-\sqrt{2}\Rightarrow$$$$f\left(x\right)\sim \frac{x\sqrt{3}}{2}(-\frac{3\sqrt{2}+2}{3x}+\frac{2\sqrt{2}}{3x^3})-\sqrt{2}$$$$\Rightarrow f\left(x\right)\sim -\frac{\sqrt{3}}{2}\left(\frac{3\sqrt{2}+2}{3}\right)+\frac{\sqrt{6}}{3x^2}-\sqrt{2}\Rightarrow$$$${lim}_{x\to +\mathrm{\infty }}f\left(x\right)=-\frac{\sqrt{3}}{6}(3\sqrt{2}+2)-\sqrt{2}$$$$at-\mathrm{\infty }{lim}_{x\to -\mathrm{\infty }}f\left(x\right)=+\mathrm{\infty }$$

Answered by jagoll last updated on 01/Mar/20

$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{{3\mathrm{x}}^2-(3\sqrt{2}+2)\mathrm{x}+2\sqrt{2}}-(\sqrt{{(\mathrm{x}\sqrt{3}+\sqrt{2})}^2}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{{3\mathrm{x}}^2-(3\sqrt{2}+2)\mathrm{x}+2\sqrt{2}}-\sqrt{{3\mathrm{x}}^2+2\sqrt{6}\mathrm{x}+2}=$$$$\frac{-(3\sqrt{2}+2)-2\sqrt{6}}{2\sqrt{3}}=\frac{-2\sqrt{6}-3\sqrt{2}-2}{2\sqrt{3}}$$$$=-\sqrt{2}-\frac{1}{2}\sqrt{6}-\frac{1}{3}\sqrt{3}$$

Commented by john santu last updated on 01/Mar/20

$$\mathrm{good}\mathrm{sir}$$

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