2-3等价无穷小在极限运算中的作用

,1,ln(1),(1)1.(0)xexxxxxx2常见的等价无穷小有：xsinxx,tanxx,1-cosx2例1.)sin1tan1(lim310xxxx求310)]1sin1tan1(1[limxxxx原式310]sin1sintan1[limxxxxx301sin1sintanlimxxxxx301cos)sin1()cos1(sinlimxxxxxxxxxxxxxcos)sin1(1cos1sinlim2021.21e原式例2).(,1)(lim,2)(lim,)(023xpxxpxxxpxpxx求且是多项式设解,2)(lim23xxxpx),(2)(23为待定系数其中可设babaxxxxp,1)(lim0xxpx又)0(~2)(23xxbaxxxxp.1,0ab从而得xxxxp232)(故例3求下列极限①xxx1)1(lim0xexx1lim)1ln(0))1ln(~1()1ln(xexxxx)1ln(lim0xx~1)1(②sin0limsinxxxeexxxxeexxxxsin1limsinsin01例4)(lim,2112sin)(1lim030xfexxfxxx求已知解2112sin)(1lim30xxexxf由0)1(lim30xxe而)12sin)(1(lim0xxfx)1(112sin)(1lim330xxxeexxf02012sin)(1lim0xxfx02sin)(lim0xxfx从而由等价无穷小的代换性质得112sin)(1lim230xxexxfxxxfx32sin)(21lim0xxxfx22sin)(lim310122sinlim0xxx由6)(lim)(lim00xfxfxx存在，且