Question

lim (sin x)^(tanx) x趋于0

Answer 1

x→0
lim (sinx)^tanx
=lim e^ln (sinx)^tanx
根据复合函数的极限运算
=e^lim ln (sinx)^tanx
现在考虑
lim ln (sinx)^tanx
=lim ln(sinx) * tanx
=lim ln(sinx) / 1/tanx
利用等价无穷小：tanx~x
=lim ln(sinx) / 1/x
该极限为∞/∞型,利用L'Hospital法则,
=lim (ln(sinx))' / (1/x)'
=lim cosx/sinx / -1/x^2
=-lim x^2 / tanx
利用等价无穷小：tanx~x
=-lim x^2/x
=0
故,原极限=e^0=1
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