`Lim _(n rarr infty)((n*sinalpha-alpha*sinn))/(n-alpha) `

lim_(n rarr0)n*sin(1/n)

lim_(n rarr0)(cos n)/(n+1)

lim_(n rarr oo)(n!)/((n+1)!-n!)

If A=[cos alpha+sin alpha sqrt(2)sin alpha-sqrt(2)sin alpha cos alpha-sin alpha], prove that =[cos n alpha+sin n alpha sqrt(2)sin n alpha-sqrt(2)sin n alpha cos n alpha-sin n alpha] for all n in N.

If alpha,beta in(-(pi)/(2),0) such that (sin alpha+sin beta)+(sin alpha)/(sin beta)=0 and (sin alpha+sin beta)(sin alpha)/(sin beta)=-1 and lambda=lim_(n rarr oo)(1+(2sin alpha)^(2n))/((2sin beta)^(2n))

If tan beta=(n sin alpha cos alpha)/(1-n sin^(2)alpha), show that tan(alpha-beta)=(1-n)tan alpha

If tan beta=(n sin alpha cos alpha)/(1-n sin^(2)alpha), show that tan(alpha-beta)=(1-n)tan alpha

(2) lim_(n rarr0) (n!sin x)/(n)

lim_(n rarr oo)(n+(-1)^(n))/(n)