用L`HOSPITAL`S RULE 來解題目?

Question

lim(cosx)^(1/X^2) X趨近於+零

問LIMIT?

可以的話請附過程．

Answer 1

(cosx)^(1/x²)=exp(ln(cosx)/x²) (換底)
而lim(x->0+) ln(cosx)/x²=lim(x->0+)-tanx/2x=-1/2 (L' Rule)
又exp(x)為連續函數,則lim(x->a) (exp(f(x))=exp(lim(x->a) f(x))
故本題=exp(-1/2)=1/√e