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Question Number 175217 by mnjuly1970 last updated on 23/Aug/22

$$$$$${lim}_{x\to 0}\{\lfloor \frac{cos\left(x\right)}{x}\rfloor -\lfloor \frac{1+cos\left(x\right)}{1-cos\left(x\right)}\rfloor \}$$

Commented by kaivan.ahmadi last updated on 24/Aug/22

$$weknowthat{\left[x\right]}_{x\to \mathrm{\infty }}\sim xso$$$$li\underset{x\to 0}{m}\frac{cosx}{x}-\frac{1+cosx}{1-cosx}=$$$$li\underset{x\to 0}{m}\frac{cosx(1-cosx)-x(1+cosx)}{x(1-cosx)}$$$$=li\underset{x\to 0}{m}\frac{(1-\frac{x^2}{2})\left(\frac{x^2}{2}\right)-x(2-\frac{x^2}{2})}{x\left(\frac{x^2}{2}\right)}$$$$=li\underset{x\to 0}{m}\frac{-2x}{\frac{x^3}{3}}=-\mathrm{\infty }$$

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