Evaluate: `("lim")_(nvecoo)ncos(pi/(4n))sin(pi/(4n))` where `(x , y)vec(0,0)` along the curve `x+y^2`

Evaluate: (lim_(n rarr oo)n cos((pi)/(4n))sin((pi)/(4n))

Evaluate: lim_(x rarr0)(y^(2)+sin x)/(x^(2)+sin y^(2)) where (x,y)rarr(0,) along the curve x=y^(2)

[sin n-cos x]_(0)^( pi/4)

cos(pi/4-x)cos(pi/4-y)-sin(pi/4-x)sin(pi/4-y)=sin(x+y)

Prove that: cos((pi)/(4)-x)cos((pi)/(4)-y)-sin((pi)/(4)-x)sin((pi)/(4)-y)=sin(x+y)

Evaluate lim_(x rarr(pi pi)/(4))[sin x+cos x], where [.] denotes the greatest integer function.

lim_(n rarr oo)(1)/(1+n sin^(2)x), then find f((pi)/(4)) and also comment on the continuity at x=0

Evaluate: int_(0)^(n pi+t)(|cos x|+|sin x|)dx, where t in[0,f[(pi)/(2))

Let f(x) be defined as f(x)=max{a,b,c} where a=lim_(n rarr oo)lim_(alpha rarr1^(+))(alpha^(n)|sin x|+|cos x| alpha^(-n))/(alpha^(n)+alpha^(-n)) , b=lim_(n rarr oo)lim_(alpha rarr1^(-))(alpha^(n)|sin x|+|cos x| alpha^(-n))/(alpha^(n)+alpha^(-n)) , c=lim_(n rarr oo)(pi)/(4n)[1+cos(pi)/(2n)+...+cos((n-1)pi)/(2n)] then The range of f(x) is

The value of lim_(n rarr oo)(1)/(n^(2)){(sin^(3)pi)/(4n)+2(sin^(3)(2 pi))/(4n)+...+n(sin^(3)(n pi))/(4n)} is equal to