11I guess, it should be zero
(1)if f'(0)=0 and f(x) is a differentiable and increasing function then
\lim_{x\rightarrow 0}\frac{xf'(x^{2})}{f'(x)}=
(a) is always 0
(b) may not exist as LHL may not exist
(c) may not exist as RHL may not exist
(d) RHL is always 0
(2) let f(r) be the no. of integral points inside a circle of radius r and centre origin(integral point is a point whose both coordinates are integers) ,then
\lim_{r\rightarrow infinity}\frac{f(r)}{\Pi r^{2}}=
-
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10 Answers
66@decoder, where did you get problem 2 from. The required limit is 1. However, I think the subject matter goes beyond the reach of an IITJEE aspirant. You can have a look at the following three links:
http://mathworld.wolfram.com/GausssCircleProblem.html
http://en.wikipedia.org/wiki/Gauss_circle_problem#cite_note-Hardy-0
http://www.math.ucdavis.edu/~latte/latex/intro/intro/
3(1)
LHL ... say x = -h
-h f/(h^2)/f/(-h)
now as h -->0
[-20^2f//(0^2) - f/(0^2)]/f//(0) = 0 !!!!
next for RHL put x = h ........ do same use l hospital ... ull get 0 again ... i guess tushar is right!!
1but if f ''(0) is also zero then it's indeterminate na.........and this will be also the case here ............. as msp pointed out x=0 is the point of inflection so f ''(0)=0......