极限题目lim(x->0)(tanx)^sinx
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极限题目
lim(x->0)(tanx)^sinx
lim(x->0)(tanx)^sinx
先计算
lim(x→0)sinx*ln(tanx)
= lim(x→0)x*ln(tanx) (等价无穷小替换)
= lim(x→0)ln(tanx)/(1/x) (0/0)
= lim(x→0){[(secx)^2]/tanx}/(-1/x^2)
= -lim(x→0)[(x^2)/tanx]*(secx)
= 0,
因此,
lim(x→0)(tanx)^sinx
= e^lim(x→0)sinx*ln(tanx)
= e^0 = 1.
lim(x→0)sinx*ln(tanx)
= lim(x→0)x*ln(tanx) (等价无穷小替换)
= lim(x→0)ln(tanx)/(1/x) (0/0)
= lim(x→0){[(secx)^2]/tanx}/(-1/x^2)
= -lim(x→0)[(x^2)/tanx]*(secx)
= 0,
因此,
lim(x→0)(tanx)^sinx
= e^lim(x→0)sinx*ln(tanx)
= e^0 = 1.
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