x^(x)-2^(sin x)

Prove that: ,,(sin x-sin3x)/(sin^(2)x-cos^(2)x)=2sin x

lim_(x rarr0)(2+2x+sin2x)/((2x+sin2x)e^(sin x)) is

lim_ (x rarr oo) (2 + 2x + sin2x) / ((2x + sin2x) e ^ (sin x)) is:

Prove that (sin x-sin 3 x)/(sin ^(2) x-cos ^(2) x)=2 sin x

the value of lim_(x rarr0)((sin2x)/(3x)+(x sin x^(2))/(sin x^(3))) is

Ascertain the existence of the following limits : lim_(xtooo) (2+2x+sin 2x)/((2x+sin 2x)e^(sin x))

lim_ (x rarr0 ^ (+)) {lim_ (n rarr oo) ([1 ^ (2) (sin x) ^ (x)] + [2 ^ (2) (sin x) ^ (x)] +. ..... * [n ^ (2) (sin x) ^ (x)]) / (n ^ (3)))}

If f(x)=det[[x,cos x,e^(x^(2))sin x,x^(2),sin^(2)xtan x,1,2]] then the value of ,f(x)dx is equal to

lim_(x rarr0)(x^(2)sin^(2)x)/(x^(2)-sin^(2)x)