Lim-x-x-4-1-cos-1-cos-2-x-

Question Number 184859 by mnjuly1970 last updated on 12/Jan/23

{Lim}_{x \to \infty} x^{4} (1 - c o s (1 - c o s (\frac{2}{x}))) = ?

Answered by qaz last updated on 12/Jan/23

{\underset{x \to \infty}{l i m x}}^{4} (1 - \cos (1 - \cos (\frac{2}{x})))

= {\underset{x \to \infty}{l i m x}}^{4} (1 - co s (\frac{2}{x^{2}} + o (\frac{1}{x^{2}}))

= {\underset{x \to \infty}{l i m x}}^{4} (\frac{2}{x^{4}} + o (\frac{1}{x^{4}})) = 2

Answered by cortano1 last updated on 13/Jan/23

\frac{1}{x} = t

\lim_{t \to 0} \frac{\sin^{2} (1 - \cos 2 t)}{2 t^{4}} = \lim_{t \to 0} \frac{\sin^{2} (2 \sin^{2} t)}{2 t^{4}}

= \frac{4}{2} = 2

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