lim-x-0-ln-sec-ex-sec-e-2-x-sec-e-50-x-e-2-e-2cos-x-

Question Number 35736 by rahul 19 last updated on 22/May/18

\lim_{x \to 0} \frac{\ln (\sec (e x) \sec (e^{2} x) \dots \dots \sec (e^{50} x))}{e^{2} - e^{2 \cos x}} = ?

Answered by tanmay.chaudhury50@gmail.com last updated on 23/May/18

t h i s l i m i t i s (\frac{0}{0}) f o r m

= \frac{l i m}{x \to 0} \frac{l n s e c (e x) + l n s e c (e^{2} x) + \dots + l n s e c (e^{50} x)}{e^{2} - e^{2 c o s x}}

u s i n g l, h o s p i t a l r u l e

= \frac{l i m}{x \to 0} \frac{t a n (e x) + t a n (e^{2} x) + . . + t a n (e^{50} x)}{0 - e^{2 c o s x} \times (- 2 s i n x)}

d e v i d i n g N_{r} a n d D_{r} b y x

= \frac{l i m}{x \to 0} \frac{e \frac{t a n (e x)}{e x} + e^{2} \frac{t a n (e^{2} x)}{e^{2} x} + . . + e^{50} \frac{t a n (e^{50} x)}{e^{50} x}}{2 e^{2 c o s x} \times \frac{s i n x}{x}}

= \frac{e + e^{2} + e^{3} + \dots + e^{50}}{2 e^{2}}

= \frac{e (\frac{e^{50} - 1}{e - 1})}{2 e^{2}}

= \frac{1}{2} \times \frac{1}{e} \times \frac{e^{50} - 1}{e - 1}

Commented by rahul 19 last updated on 25/May/18

Thank You Sir . There is a slight error in 2nd line .�� It will be e*tan(ex) ,......