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lim 1 x tAnx

lim (1/x)^tanx 根据等价无穷小简化成 lim (1/x)^x 【x→0+】 =lim 1/ x^x 对x^x取对数lnx^x,得xlnx,化成lnx / [1/x] 洛必达法则: 上下求导,分子1/x 分母-1/x^2 结果= -x 所以极限lnx^x= -x=0 那么x^x的极限就是e^0=1 所以lim (1/x)^tanx =li...

解:lim(x->1)(1-x)tan(πx/2) =lim(y->0)[y*tan(π/2-πy/2)] (用y=1-x代换) =lim(y->0)[y*ctan(πy/2)] =lim(y->0)[y*cos(πy/2)/sin(πy/2)] =lim(y->0){[(πy/2)/sin(πy/2)]*[(2/π)*cos(πy/2)]} ={lim(y->0)[(πy/2)/sin(πy/2)]}*{lim(y->0)[(2/π)*c...

设y=(1/x)^tanx= lny=tanx*ln(1/x) lim0> lny=lim tanx*ln(1/x)=lim ln(1/x)/ctanx=lim (-1/x)/(-csc²x)=lim sin²x/x=lim sinx/x * sinx=1*0=0 lim0>lny=0 所以 lim(1/x)∧tanx=e^0=1

原式=lim(x->0)[(1-cosx)/(xsinx)] =lim(x->0)[(0.5x^2)/x^2] =0.5

tanx/x = sinx /(xcosx)=1/ cosx =1

解: lim ln[(1/x)^(tanx)] x→0 =lim tanx·ln[(1/x) x→0 =lim ln[(1/x)/cotx x→0 =lim x·(-1/x²)/(-csc²x) x→0 =lim sin²x/x x→0 =lim 2sinxcosx/1 x→0 =lim sin2x x→0 =sin0 =0 lim [(1/x)^(tanx)] =e^0 =1 x→0 对于不方便直接...

tanx ~ x ln (1-x) ~ -x 原来极限=x^(-x) = e^(-x lnx ) xlnx = lnx /(1/x) => 1/x /(-1/x^2) = -x 所以原来极限=1

先把指数等价替换成1/x,再用两种方法(省略极限符号): 方法一:e^[ln(1+x)-ln(1-x)]/x =e^[x+o(x)-(-x)+o(x)]/x =e^[2x+o(x)]/x =e² 方法二:{[1+2x/(1-x)]^[(1-x)/2x]}^[2/(1-x)] =e^(2/1) =e²

解析如图

lim(x→0)(tanx/x^3-1/x^2) =lim(x→0) (tanx - x)/x^3 (0/0分子分母分别求导) =lim(x→0) [(secx)^2 - 1]/(3x^2) =lim(x→0) [ 1- (cosx)^2]/[ (3x^2).(cosx)^2] =lim(x→0) (sinx)^2/ (3x^2) =1/3 or x->0 tanx = x+(1/3)x^3 +o(x^3) lim(x→0)(...

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