Which of the following true ({.} denotes the fractional part of the function)? `("lim")_(xvecoo)((log)_e x)/({x})=oo` (b) `("lim")_(xvec2^+)x/(x^2-x-2)=oo` `("lim")_(xvec1^-)x/(x^2-x-2)=-oo` (d) `("lim")_(xvecoo)((log)_(0.5) x)/({x})=oo`

Given a real-valued function f such that f(x)={(tan^2{x})/((x^2-[x]^2) sqrt({x}cot{x}),for x 0 Where [x] is the integral part and {x} is the fractional part of x , then ("lim")_(xvec0^+)f(x)=1 , ("lim")_(xvec0^-)f(x)=cot1 , cot^(-1)(("lim")_(xvec0^-)f(x))^2=1 , tan^(-1)(("lim")_(xvec0^+)f(x))=pi/4

lim_(x->0)log(1+2x)/x

Using lim_(x to 0)(e^(x)-1)/(x)=1 , show that, lim_(x to 0)log_(e)(1+x)/(x)=1

lim_(x to oo) sinx/x = ?

If f(x)=(cosx)/((1-sinx)^(1/3)) then (a) ("lim")_(xrarrpi/2)f(x)=-oo (b) ("lim")_(xrarrpi/2)f(x)=oo (c) ("lim")_(xrarrpi/2)f(x)=0 (d) none of these

Column I ([.] denotes the greatest integer function), Column II ""("lim")_(xvec0)([100(sinx)/x]+[100(tanx)/x]) , p. 198 ("lim")_(xvec0)([100 x/(sinx)]+[100(tanx)/x]) , q. 199 ("lim")_(xvec0)([100(sin^(-1)x)/x]+[100(tan^(-1)x)/x]) , r. 200 ("lim")_(xvec0)([100 x/(sin^(-1)x)]+[100(tan^(-1)x)/x]) , s. 201

If f^(prime)(x)=|x|-{x}, where {x} denotes the fractional part of x , then f(x) is decreasing in (a) (-1/2,0) (b) (-1/2,2) (-1/2,2) (d) (1/2,oo)

Discuss the continuity of the function, f(x)=lim_(nrarrinfty)(log(2+x)-x^(2n)sinx)/(1+x^(2n)) at x=1

lim_(x->oo)[x-log_e((e^x+e^(-x))/2)]= a) (log)_e4 b. 0 c. oo d. (log)_e2

Maximum value of (log x)/(x) in [2, oo) is-