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Question Number 144558 by imjagoll last updated on 26/Jun/21

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

Answered by liberty last updated on 26/Jun/21

$$\underset{x\to -\mathrm{\infty }}{\lim }\left(\frac{2\mathrm{x}-\sqrt{{3\mathrm{x}}^2+3\mathrm{x}}}{\mathrm{x}-1}\right)=?$$$$\mathrm{Sol}:$$$$\underset{x\to -\mathrm{\infty }}{\lim }\left(\frac{2\mathrm{x}-\sqrt{{3\mathrm{x}}^2+3\mathrm{x}}}{\mathrm{x}-1}\right)$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\left(\frac{2\mathrm{x}-\mid \mathrm{x}\mid \sqrt{3+\frac{3}{\mathrm{x}}}}{\mathrm{x}-1}\right)$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{\mathrm{x}(2+\sqrt{3+{3\mathrm{x}}^{-1}}}{\mathrm{x}(1-{\mathrm{x}}^{-1})}$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{2+\sqrt{3+{3\mathrm{x}}^{-1}}}{1-{\mathrm{x}}^{-1}}$$$$=2+\sqrt{3}$$

Answered by Olaf_Thorendsen last updated on 26/Jun/21

$$f\left(x\right)=\frac{2x-\sqrt{3x^2+3x}}{x-1}$$$$f\left(x\right)=\frac{4x^2-(3x^2+3x)}{(x-1)(2x+\sqrt{3x^2+3x})}$$$$f\left(x\right)=\frac{x^2-3x}{(x-1)(2x+\sqrt{3x^2+3x})}$$$$f\left(x\right)=\frac{x^2-3x}{(x-1)(2x+\mid x\mid \sqrt{3+\frac{3}{x}})}$$$$\underset{x\to -\mathrm{\infty }}{\lim }f\left(x\right)$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{x^2-3x}{(x-1)(2x-x\sqrt{3+\frac{3}{x}})}$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{x-3}{(x-1)(2-\sqrt{3+\frac{3}{x}})}$$$$=\frac{1}{2-\sqrt{3}}=2+\sqrt{3}$$

Answered by mathmax by abdo last updated on 26/Jun/21

$$={\lim }_{\mathrm{x}\to -\mathrm{\infty }}(\frac{2\mathrm{x}}{\mathrm{x}-1}-\frac{\mid \mathrm{x}\mid \sqrt{3+\frac{3}{\mathrm{x}}}}{\mathrm{x}-1})$$$$=2+{\lim }_{\mathrm{x}\to -\mathrm{\infty }}\frac{\mathrm{x}}{\mathrm{x}-1}\sqrt{3+\frac{3}{\mathrm{x}}}=2+\sqrt{3}$$

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