lim-x-1-sec-pi-2-x-arctanx-pi-4-

Question Number 215667 by mathlove last updated on 14/Jan/25

\lim_{x \to 1} s e c (\frac{π}{2} x) (a r c t a n x - \frac{π}{4}) = ?

Answered by issac last updated on 14/Jan/25

\sec (\frac{π}{2}) (\arctan (1) - \frac{π}{4})

Commented by Ghisom last updated on 14/Jan/25

just tell us the value of \sec \frac{π}{2}

Commented by mathlove last updated on 14/Jan/25

c o r e c t a n s w e r i s - \frac{1}{π}

Answered by Ghisom last updated on 14/Jan/25

\lim_{x \to 1} (\arctan x - \frac{π}{4}) \sec \frac{π x}{2} =

= \lim_{x \to 1} \frac{\arctan x - \frac{π}{4}}{\cos \frac{π x}{2}} =

[l^{'} H \hat{o p i t a l}]

= \lim_{x \to 1} \frac{\frac{1}{x^{2} + 1}}{- \frac{π}{2} \sin \frac{π x}{2}} = \frac{\frac{1}{2}}{- \frac{π}{2}} = - \frac{1}{π}

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