If `f(x)=lim_(t->oo)((|alpha+sin pi x|)^t-1)/((|alpha+sinpi x|)^t+1) , x in (0,6)` then

lim_(x rarr oo)(sin alpha-alpha sin x)/(x-alpha)

lim_(x rarr0)(sin(alpha+x)-sin(alpha-x))/(cos(alpha+x)-cos(alpha-x))

If f(x)=lim_(t to 0)[(2x)/(pi).tan^(-1)(x/(t^(2)))] ,then f(1) is …….

If f(x)=lim_(t to 0)[(2x)/(pi).tan^(-1)(x/(t^(2)))] ,then f(1) is …….

If f(x)=lim_(t to 0)[(2x)/(pi).tan^(-1)(x/(t^(2)))] ,then f(1) is …….

If f(x)=lim_(t to 0)[(2x)/(pi).tan^(-1)(x/(t^(2)))] ,then f(1) is …….

f(x)=lim_(n rarr oo)(tan pi x^(2)+(x+1)^(n)sin x)/(x^(2)+(x+1)^(n)), find lim_(x rarr0)f(x)

f(x)=int_(0)^(x)(e^(t)-1)(t-1)(sin t-cos t)sin tdt,AA x in(-(pi)/(2),2 pi) then f(x) is decreasing in :

Let f:(0, pi)->R be a twice differentiable function such that (lim)_(t->x)(f(x)sint-f(x)sinx)/(t-x)=sin^2x for all x in (0, pi) . If f(pi/6)=-pi/(12) , then which of the following statement(s) is (are) TRUE? f(pi/4)=pi/(4sqrt(2)) (b) f(x)<(x^4)/6-x^2 for all x in (0, pi) (c) There exists alpha in (0, pi) such that f^(prime)(alpha)=0 (d) f"(pi/2)+f(pi/2)=0