大一高数求详解
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cos x ~ 1-(1/2)x^2
所以 (cos x)^(2/(x^2)) = exp(ln(cosx)·2/(x^2))
~exp(ln(1-(1/2)x^2)·2/(x^2))
~exp(-(1/2)*(x^2)*2/(x^2))
=exp(-1)=e^(-1)
所以 (cos x)^(2/(x^2)) = exp(ln(cosx)·2/(x^2))
~exp(ln(1-(1/2)x^2)·2/(x^2))
~exp(-(1/2)*(x^2)*2/(x^2))
=exp(-1)=e^(-1)
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lim(x→0) (cosx)^(2/x^2)
=lim(x→0) [1-2sin^2(x/2)]^(2/x^2)
=lim(x→0) [1-2sin^2(x/2)]^{1/[2sin^2(x/2)]*4sin^2(x/2)*1/x^2}
=lim(x→0) [1-2sin^2(x/2)]^{1/[2sin^2(x/2)]}
=1/e
=lim(x→0) [1-2sin^2(x/2)]^(2/x^2)
=lim(x→0) [1-2sin^2(x/2)]^{1/[2sin^2(x/2)]*4sin^2(x/2)*1/x^2}
=lim(x→0) [1-2sin^2(x/2)]^{1/[2sin^2(x/2)]}
=1/e
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