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Question Number 114088 by bemath last updated on 17/Sep/20

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{2x\cot \left(\frac{2}{x}\right)-3\cot \left(\frac{2}{x}\right)}{5x^2-2x}$$

Answered by Olaf last updated on 17/Sep/20

$$=\underset{x\to \mathrm{\infty }}{\lim }\frac{1-\frac{1}{2}{\left(\frac{2}{x}\right)}^2}{\left(\frac{2}{x}\right)}\left[\frac{2x-3}{5x^2-2x}\right]$$$$=\underset{x\to \mathrm{\infty }}{\lim }\frac{x^2-2}{2x}\times \frac{2x-3}{5x^2-2x}$$$$=\underset{x\to \mathrm{\infty }}{\lim }\frac{2x^3}{10x^3}=\frac{1}{5}$$

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