calculate-lim-x-0-arctan-e-x-pi-4-x-2-

Question Number 72990 by mathmax by abdo last updated on 05/Nov/19

c a l c u l a t e {l i m}_{x \to 0} \frac{a r c t a n (e^{x}) - \frac{π}{4}}{x^{2}}

Commented by abdomathmax last updated on 19/Nov/19

l e t u s e h o s p i t a l t h e o r e m f (x) = a r c t a n (e^{x}) - \frac{π}{4} a n d

g (x) = x^{2} w e h a v e f^{^{'}} (x) = \frac{e^{x}}{1 + e^{2 x}}

f^{(2)} (x) = \frac{e^{x} (1 + e^{2 x}) - 2 e^{2 x} e^{x}}{{(1 + e^{2 x})}^{2}} = \frac{e^{x} + e^{3 x} - 2 e^{3 x}}{{(1 + e^{2 x})}^{2}}

= \frac{e^{x} - e^{3 x}}{{(1 + e^{2 x})}^{2}} \Rightarrow {l i m}_{x \to 0} f^{(2)} (x) = 0

g^{^{'}} (x) = 2 x a n d g^{(2)} (x) = 2 \Rightarrow {l i m}_{x \to 0} g^{(2)} (x) = 2 \Rightarrow

{l i m}_{x \to 0} \frac{a r c t a n (e^{x}) - \frac{π}{4}}{x^{2}} = 0

Answered by mind is power last updated on 05/Nov/19

= \lim x \to 0 \frac{\frac{e^{x}}{1 + e^{2 x}}}{2 x} = {\begin{cases} + \infty x \to 0^{+} \\ - \infty x \to 0^{-} \end{cases}