两个重要极限及其应用

sin1lim1,lim(1)0xxxxx=+=→→∞sin1lim1,lim(1)xexxxx=+=→→∞sinlim10xxx=→lim1sin0()sin()lim1,lim1sin()()()0()0xxxxxxxxxjjjjjj=→==→→()xj000sin2limxxx→00sin2sin2limlim222xxxxxx→→=×=lim2sin(0)2nnnxx→∞≠sin2lim2sin(0)lim22nnnnnnxxxxxx→∞→∞≠=×=sinlimcosxxxxpp→-sinsin()limcoslim[]coslimcos1xxxxxxxxxxpppppp→→→-=-=-=--()0xj→()lim1sin()()0xxxjjj=→()0xj→0lim()0,lim()0,lim()0xxaxxxxjjj→→→∞===1lim(1)xexx+=→∞1()1.lim(1()),()0xxexjjj+=→12lim()xxxx+→∞22112lim()lim[(1)]exxxxxxx=+=+→∞→∞2lim(cos)0xxx-→2(cos1)22cos1lim(cos)lim[1(cos1)]lim[1(cos1)]0002(cos1)()(cos1)0{lim[1(cos1)]}10xxxxxxxxxxxxxxxex-×----=+-=+-→→→---=+-==→10lim()(0,0,0)3xxxxxabcabc→++11100033lim()lim(1)lim(1)333xxxxxxxxxxxxxxxabcabcabc→→→++++-++-=+=+0033330311111limlim()(lnlnln)33333lim(1)3xxxxxxxxxxxxxxabcxxxxabcxabcabcabcxxxxabceeeabc→→++-×++-→++----++++++-=+====()0xj→0lim()0,lim()0,lim()0xxaxxxxjjj→→→∞===lim()0,lim()0,lim()0xxaxxxxjjj→→→∞===1()2.lim(1)()()xexxjjj+=→∞xxxx)3212(lim++∞→xxxxxxx)2111(1)1221(1)3212(++=++=++xxx)2111(lim++∞→eexxxx=⋅=++⋅++=-+∞→1)2111()2111(lim2121xxxx)3212(lim++∞→e1=0)(lim=→AxfaxBxfax=→)(limBxgaxAxf=→)()(lim1322232])3221[()3221()3212(-+-+--→+-+=+-+=++exxxxxxxxx()xj→∞0lim(),lim(),lim()xxaxxxxjjj→→→∞=∞=∞=∞0x→x→∞sin1lim1,lim(1)0xxexxxx=+=→∞→