Find the sum of an infinite geometric series whose first term is the limit of the function `f( x) =( tan x - sinx )/( sin^(3) x ) ` as `x rarr 0 ` and whose common ratio is the limit of the function `g(x) = ( 1-sqrt(x))/( (cos^(-1)x )` as `x rarr 1 `

Find the sum of an infinite geometric series whose first term is the limit of the function f(x)=(tan x-sin x)/(sin^(3)x) as x rarr0 and whose common ratio is the limit of the function g(x)=(1-sqrt(x))/((cos^(-1)x)^(2)) as x->1

Find the sum of an infinite geometric series whose first term is the limit of the function f(x)=(1-tan x)/(1-sqrt(2)sin x) as x rarr(pi)/(4) and whose common ratio is the limit of the function g(x)=(1-sqrt(x))/((cos^(-1)x)^(2)) as x rarr1

Let S denotes the sum of an infinite geometric progression whose first term is the value of the functions f(x)=(sin(x-pi//6))/(sqrt3-2cosx) at x=pi/6, if f(x) is continuous at x=pi/6 and whose common ratio is the limiting value of the function g(x)=(sin(x)^(1//3) ln(1+3x))/((arc tansqrtx)^2 (e^(5*x^(1//3))-1)) as x rarr 0. Find the value of 2S.

Find the sum of an infinite geometric series whose first term is lim_(x rarr0)sum_(k=1)^(2011)(((x)/(tan x)+2k))/(2011) and whose common ratio is the value of lim_(x rarr0)(e^(tan^(3)x)-e^(x^(3)))/(2ln(1+x^(3)sin^(2)x))[ Note: (y) denotes fractional part of yl.

The limit of x sin((1)/(e^(x))) as x rarr 0

The limit of x sin (e^(-1//x)) as x rarr0

The limit of [(1)/(x)sqrt(1+x)-sqrt(1+(1)/(x^(2))]] as x rarr0

The limit of {(1)/(x)sqrt(1+x) - sqrt(1+(1)/(x^(2))} as x rarr 0

The limiting value of x sin((1)/(x)) as x rarr0 is

Evaluate the limit: lim_(x rarr0)(sqrt(2)-sqrt(1+cos x))/(sin^(2)x)