lim(x→0)(cosx)^(1/ln(1+x^2))

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lim(x→0)(cosx)^(1/ln(1+x^2))
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lim(x→0)(cosx)^(1/ln(1+x^2))
lim(x→0)(cosx)^(1/ln(1+x^2))

lim(x→0)(cosx)^(1/ln(1+x^2))
设 f(x) = (cosx)^(1/ln(1+x^2)),ln f(x) = ln(cosx) / ln(1+x^2)
x→0,ln(cosx) = ln [1 + (cosx - 1] cosx - 1 - x^2 /2
ln(1+x^2) x^2
lim(x→0) ln(cosx) / ln(1+x^2)
= lim(x→0) (-1/2) x^2 / x^2 = -1/2
lim(x→0)(cosx)^(1/ln(1+x^2)) = e^(-1/2)

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