lim(n→∞) (cos x/n)^n^2

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lim(n→∞) (cos x/n)^n^2

lim(n→∞) (cos x/n)^n^2
lim(n→∞) (cos x/n)^n^2

lim(n→∞) (cos x/n)^n^2
lim(n→∞) (cos x/n)^n^2=?
y=[cos(x/n)]^n² lny = lncos(x/n) / (1/n²)
lim(n->wq) lny = -(-x/n²) sin(x/n)/cos(x/n) / (-2/n³)
= -(x/2) lim(n->wq) n tan(x/n)
= -(x/2) lim(n->wq) sec²(x/n) (-x/n²)/(-1/n²)
= -(x²/2) lim(n->wq) sec²(x/n)
= -x²/2
因此:
lim(n→∞) (cos x/n)^n^2 = e^(-x²/2)