lim-x-csc-2-2-x-1-4-x-2-

Question Number 112917 by abdullahquwatan last updated on 10/Sep/20

\lim_{x \to \infty} [\csc^{2} (\frac{2}{x}) - \frac{1}{4} x^{2}]

Commented by bobhans last updated on 10/Sep/20

\lim_{w \to 0} (\frac{2 w + \sin (2 w)}{4 w}) (\frac{2 w - \sin (2 w)}{w . \sin^{2} (2 w)})

1 \times \lim_{w \to 0} (\frac{2 w - \sin 2 w}{{4 w}^{3}}) =

\lim_{w \to 0} \frac{2 - 2 \cos 2 w}{{12 w}^{2}} = \lim_{w \to 0} (\frac{1 - \cos 2 w}{{6 w}^{2}})

= \lim_{w \to 0} (\frac{2 \sin^{2} w}{{6 w}^{2}}) = \frac{1}{3}

Commented by abdullahquwatan last updated on 10/Sep/20

thank you sir

Commented by abdullahquwatan last updated on 10/Sep/20

why {4 w}^{3} ? ?

Answered by mathmax by abdo last updated on 10/Sep/20

let f (x) = \frac{1}{\sin^{2} (\frac{2}{x})} - \frac{x^{2}}{4} changement \frac{2}{x} = t give \frac{x}{2} = \frac{1}{t} \Rightarrow x = \frac{2}{t}

(x \to \infty \Leftrightarrow t \to 0) \Rightarrow f (x) = \frac{1}{\sin^{2} t} - \frac{1}{4} (\frac{4}{t^{2}}) = \frac{1}{\sin^{2} t} - \frac{1}{t^{2}} = g (t)

= \frac{t^{2} - \sin^{2} t}{t^{2} \sin^{2} t} we have sint \sim t - \frac{t^{3}}{6} \Rightarrow \sin^{2} t \sim {(t - \frac{t^{3}}{6})}^{2} = t^{2} - \frac{1}{3} t^{4} + \frac{t^{6}}{36}

\sim t^{2} - \frac{1}{3} t^{4} \Rightarrow g (t) \sim \frac{t^{2} - t^{2} + \frac{1}{3} t^{4}}{t^{2} . t^{2}} = \frac{1}{3} \Rightarrow \lim_{t \to 0} g (t) = \frac{1}{3} \Rightarrow

\lim_{x \to \infty} f (x) = \frac{1}{3}

Commented by abdullahquwatan last updated on 10/Sep/20

thank you sir

Commented by mathmax by abdo last updated on 11/Sep/20

you are welcome