lim-x-x-2-sec-2-x-1-

Question Number 12513 by Joel577 last updated on 24/Apr/17

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

Answered by ajfour last updated on 24/Apr/17

= \lim_{x \to \infty} x^{2} [\frac{1}{\cos (2 / x)} - 1]

= \lim_{x \to \infty} x^{2} [\frac{1 - \cos (2 / x)}{\cos (2 / x)}]

= \lim_{x \to \infty} x^{2} [\frac{2 \sin^{2} (1 / x)}{\cos (2 / x)}]

= \lim_{x \to \infty} [(2) {(\frac{\sin (1 / x)}{(1 / x)})}^{2} . \frac{1}{\cos (2 / x)}]

= 2 \times 1 \times \frac{1}{1} = 2 .

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