Question

lim x→0+ x次根号下cos 根号下x等于多少,

Answer 1

x→0+
lim (cos√x)^(1/x)
=lim e^ln (cos√x)^(1/x)
=e^[lim ln (cos√x)^(1/x)]
现在考虑
lim ln (cos√x)^(1/x)
=lim ln(cos√x) / x
=lim ln(1+(cos√x-1)) / x
利用等价无穷小：ln(1+x)~x
=lim (cos√x-1) / x
=-lim (1-cos√x) / x
利用等价无穷小：1-cosx~x^2/2
=-lim x/2 / x
=-lim 1/2
=-1/2
因此,原极限=e^(-1/2)
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