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Question Number 128992 by liberty last updated on 12/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }(\mathrm{x}\sqrt{\mathrm{x}-4}-\sqrt{{\mathrm{x}}^3+5\mathrm{x}})=?$$

Answered by bramlexs22 last updated on 12/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{{\mathrm{x}}^3-{4\mathrm{x}}^2}-\sqrt{{\mathrm{x}}^3+5\mathrm{x}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{-{4\mathrm{x}}^2-5\mathrm{x}}{\sqrt{{\mathrm{x}}^3-{4\mathrm{x}}^2}+\sqrt{{\mathrm{x}}^3+5\mathrm{x}}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{-{4\mathrm{x}}^2-5\mathrm{x}}{{\mathrm{x}}^2(\sqrt{\frac{1}{\mathrm{x}}-\frac{4}{{\mathrm{x}}^2}}+\sqrt{\frac{1}{\mathrm{x}}+\frac{5}{{\mathrm{x}}^3}})}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{-4-\frac{5}{\mathrm{x}}}{\sqrt{{\mathrm{x}}^{-1}-{4\mathrm{x}}^{-2}}+\sqrt{{\mathrm{x}}^{-1}+{5\mathrm{x}}^{-3}}}=-\mathrm{\infty }$$

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