lim-x-cos-2-x-x-1-2x-

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

\lim_{x \to \infty} \frac{\cos^{2} x - x}{1 - 2 x}

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

l e t f (x) = \frac{{c o s}^{2} x - x}{1 - 2 x} \Rightarrow f o r x \neq 0 w e h a v e f (x) = \frac{x - {c o s}^{2} x}{2 x - 1}

= \frac{x (1 - \frac{{c o s}^{2} x}{x})}{x (2 - \frac{1}{x})} = \frac{1 - \frac{{c o s}^{2} x}{x}}{2 - \frac{1}{x}} b u t {l i m}_{x \to + \infty} \frac{{c o s}^{2} x}{x} = 0 b e c s u s e ∣ c o s x ∣⩽ 1

{l i m}_{x \to + \infty} \frac{1}{x} = 0 \Rightarrow {l i m}_{x \to + \infty} f (x) = \frac{1}{2}

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

\equiv {l i m}_{x \to \infty} \frac{- x}{- 2 x} = \frac{1}{2}