求极限中无穷小代换和高阶无穷小略去问题Lim (1/x^2-cot^2x)=lim (1/x^2-1/tan^2x)=lim(tan^2x-x^2)/x^2*tan^2x=lim(tan^2x-x^2)/x^4=lim(tan^2x/x^4)=lim2tanxsec^2x/4x^3=lim2xsec^2x/4x^3=limsec^2/2x^2=lim2sec^2xtanx/4x=limsec^2x*x/2x=limsec
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![求极限中无穷小代换和高阶无穷小略去问题Lim (1/x^2-cot^2x)=lim (1/x^2-1/tan^2x)=lim(tan^2x-x^2)/x^2*tan^2x=lim(tan^2x-x^2)/x^4=lim(tan^2x/x^4)=lim2tanxsec^2x/4x^3=lim2xsec^2x/4x^3=limsec^2/2x^2=lim2sec^2xtanx/4x=limsec^2x*x/2x=limsec](/uploads/image/z/3654888-24-8.jpg?t=%E6%B1%82%E6%9E%81%E9%99%90%E4%B8%AD%E6%97%A0%E7%A9%B7%E5%B0%8F%E4%BB%A3%E6%8D%A2%E5%92%8C%E9%AB%98%E9%98%B6%E6%97%A0%E7%A9%B7%E5%B0%8F%E7%95%A5%E5%8E%BB%E9%97%AE%E9%A2%98Lim+%281%2Fx%5E2-cot%5E2x%29%3Dlim+%281%2Fx%5E2-1%2Ftan%5E2x%29%3Dlim%28tan%5E2x-x%5E2%29%2Fx%5E2%2Atan%5E2x%3Dlim%28tan%5E2x-x%5E2%29%2Fx%5E4%3Dlim%28tan%5E2x%2Fx%5E4%29%3Dlim2tanxsec%5E2x%2F4x%5E3%3Dlim2xsec%5E2x%2F4x%5E3%3Dlimsec%5E2%2F2x%5E2%3Dlim2sec%5E2xtanx%2F4x%3Dlimsec%5E2x%2Ax%2F2x%3Dlimsec)
求极限中无穷小代换和高阶无穷小略去问题Lim (1/x^2-cot^2x)=lim (1/x^2-1/tan^2x)=lim(tan^2x-x^2)/x^2*tan^2x=lim(tan^2x-x^2)/x^4=lim(tan^2x/x^4)=lim2tanxsec^2x/4x^3=lim2xsec^2x/4x^3=limsec^2/2x^2=lim2sec^2xtanx/4x=limsec^2x*x/2x=limsec
求极限中无穷小代换和高阶无穷小略去问题
Lim (1/x^2-cot^2x)=lim (1/x^2-1/tan^2x)=lim(tan^2x-x^2)/x^2*tan^2x=lim(tan^2x-x^2)/x^4
=lim(tan^2x/x^4)=lim2tanxsec^2x/4x^3=lim2xsec^2x/4x^3=limsec^2/2x^2=lim2sec^2xtanx/4x=limsec^2x*x/2x=limsec^2x/2=1/2 (x趋向于0)
题目中由于x^2是高阶无穷小略去
求极限中无穷小代换和高阶无穷小略去问题Lim (1/x^2-cot^2x)=lim (1/x^2-1/tan^2x)=lim(tan^2x-x^2)/x^2*tan^2x=lim(tan^2x-x^2)/x^4=lim(tan^2x/x^4)=lim2tanxsec^2x/4x^3=lim2xsec^2x/4x^3=limsec^2/2x^2=lim2sec^2xtanx/4x=limsec^2x*x/2x=limsec
lim(tan^2x-x^2)/x^4=lim(tan^2x/x^4)
这一步错了
无穷小不能这么略去
因为下面还有一个分母x^4
如果是lim(2+x^2),x趋向于0就可以直接略去=2
但是那个你相当于略去了
lim(-x^2)/x^4=-lim1/x^2=-无穷大,所以错误.