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Question Number 107372 by bobhans last updated on 10/Aug/20

$$⋗\mathcal{B}\mathbf{obhans}⋖$$$$\underset{x\to \mathrm{\infty }}{\lim }\mathrm{x}\sqrt{\frac{2\mathrm{x}-\cos 5\mathrm{x}}{{3\mathrm{x}}^3}}?$$

Answered by john santu last updated on 10/Aug/20

$$\divideontimes \mathcal{JS}\divideontimes$$$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{x^2}\sqrt{\frac{2x-\cos 5x}{3x^3}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{\frac{2x^3-x^2\cos 5x}{3x^3}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt{\frac{2}{3}-0}=\frac{\sqrt{6}}{3}$$

Answered by Dwaipayan Shikari last updated on 10/Aug/20

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

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