21somebody take a look...
1) prove \lim_{x\rightarrow 0} sin x / x =1 by squeeze theorem.
2)prove that all roots of e^x [ d^n/d x^n(x^n/e^x)] are positive
3) prove that e is irrational.
-
UP 0 DOWN 0 0 5
5 Answers
3413) Post #7 in http://targetiit.com/iit-jee-forum/posts/limit-question-18255.html
3411) Is in the NCERT text book as far as i can remember. They consider the unit circle with centre O and a ray OA in the 1st quadrant that makes an angle of x with the +ve x-axis OB. Let AC be the projection of OA on OB. Let the tangent at A intersect OB at D
Then
OA = OB=1
AC = AC = sin x
AD = tan x
We have Area of triangle OAB< Area of sector OAB < Area of triangle OAD
Hence \sin x \cos x < x < \tan x or
\cos x < \frac{x}{\sin x} < \sec x
By squeeze principle the required limit is seen to be 1.
21thanks sir...(i was a fool ignoring it and making it by expansion series:P)
i had a proof in mind for the 1st one(generated by back calculation though)... check if it suffice
if x lies between 0 and pi/2
we always have (observation)
sinx<x<tanx
dividing by sinx we get the limit as 1