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Question Number 162182 by bobhans last updated on 27/Dec/21

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{{(\mathrm{x}-\sqrt{{\mathrm{x}}^2-3\mathrm{x}+2})}^{2022}+{(\mathrm{x}-\sqrt{{\mathrm{x}}^2-5})}^{2022}}{{\mathrm{x}}^{2022}}=?$$

Answered by cortano last updated on 27/Dec/21

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{{(x-x\sqrt{1-3x^{-1}+2x^{-2}})}^{2022}+{(x-x\sqrt{1-5x^{-2}})}^{2022}}{x^{2022}}$$$$=\underset{x\to \mathrm{\infty }}{\lim }\frac{{(1-\sqrt{1-3x^{-1}+2x^{-2}})}^{2022}+{(1-\sqrt{1-5x^{-2}})}^{2022}}{1}$$$$=0$$

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