試化簡下列極限: lim a->0 {[(x+a)*sin(a)-x*sin(x)]/[(x+a)*cos(x+a)-x*cos(x)]}?

lim a->0 {[(x+a)*sin(x+a)-x*sin(x)]/[(x+a)*cos(x+a)-x*cos(x)]}
=lim a->0 {[(x+a)*sin(x+a)-x*sin(x)](x+a-x)}/{[(x+a)*cos(x+a)-x*cos(x)]/(x+a-x)}
=lim a->0 [ d ( x sin x )/dx]/[d( x cos x )/dx]
=lim a->0 (sin x + x cos x)/(cos x - x sin x)
=(sin x + x cos x)/(cos x - x sin x)

更新 2: 延伸:
lim a->0 {[(x+a)^n*sin(x+a)-x^n*sin(x)]/[(x+a)^n*cos(x+a)-x^n*cos(x)]}
= lim a->0 [d(x^n sin x)/dx]/[d(x^n cos x)/dx ]
= [ n x^(n-1) sin x + x^n cos x ]/[ n x^(n-1) cos x - x^n sin x]
= ( n sin x + x cos x)/(n cos x - x sin x)