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Question Number 66434 by iklima_0412 last updated on 15/Aug/19

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{{\cos }^2\mathrm{x}-\mathrm{x}}{1-2\mathrm{x}}$$

Commented by mathmax by abdo last updated on 15/Aug/19

$$letf\left(x\right)=\frac{{cos}^{2}x-x}{1-2x}\Rightarrow forx\ne 0wehavef\left(x\right)=\frac{x-{cos}^{2}x}{2x-1}$$$$=\frac{x(1-\frac{{cos}^{2}x}{x})}{x(2-\frac{1}{x})}=\frac{1-\frac{{cos}^{2}x}{x}}{2-\frac{1}{x}}but{lim}_{x\to +\mathrm{\infty }}\frac{{cos}^{2}x}{x}=0becsuse\mid cosx\mid ⩽1$$$${lim}_{x\to +\mathrm{\infty }}\frac{1}{x}=0\Rightarrow {lim}_{x\to +\mathrm{\infty }}f\left(x\right)=\frac{1}{2}$$

Commented by kaivan.ahmadi last updated on 15/Aug/19

$$\equiv {lim}_{x\to \mathrm{\infty }}\frac{-x}{-2x}=\frac{1}{2}$$

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