搜搜做题作业网求极限,lim(x->0) (e^x-e^sinx ) / [ (tanx )^2 * ln(1+2x)]
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求极限,lim(x->0) (e^x-e^sinx ) / [ (tanx )^2 * ln(1+2x)]
![求极限,lim(x->0) (e^x-e^sinx ) / [ (tanx )^2 * ln(1+2x)]](/uploads/image/z/4100346-18-6.jpg?t=%E6%B1%82%E6%9E%81%E9%99%90%2Clim%28x-%3E0%29+%28e%5Ex-e%5Esinx+%29+%2F+%5B+%28tanx+%29%5E2+%2A+ln%281%2B2x%29%5D)
利用等价无穷小和L'Hospital's Rule 即可
lim(x->0) (e^x-e^sinx ) / [ (tanx )^2 * ln(1+2x)]
=lim(x->0) e^x(e^(x-sinx)-1 ) / [ (tanx )^2 * ln(1+2x)]
=lim(x->0) (x-sinx ) / [ (x)^2 * 2x)]
=lim(x->0) (1-cosx)/(6x^2)
=lim(x->0) [(x^2)/2]/(6x^2)
=1/12
lim(x->0) (e^x-e^sinx ) / [ (tanx )^2 * ln(1+2x)]
=lim(x->0) e^x(e^(x-sinx)-1 ) / [ (tanx )^2 * ln(1+2x)]
=lim(x->0) (x-sinx ) / [ (x)^2 * 2x)]
=lim(x->0) (1-cosx)/(6x^2)
=lim(x->0) [(x^2)/2]/(6x^2)
=1/12
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