设有数列An,Bn 若Bn>0递增且有n-->+∞时Bn-->+∞(以下lim均表示lim(n-->+∞)) 则有: 若lim(A(n+1)-An)/(B(n+1)-Bn)=L(L可以是0,有限数,或+∞(-∞))==>lim(An)/(Bn)=L.设有数列An,Bn若Bn0递增且有n→+∞时Bn→+∞(以下lim均表示limn→∞)则有:若lim(An+1-An)(Bn+1-Bn)=L(L可以是0,有限数,或+∞(-∞))→lim(An)(Bn)=L1)当L=0时由条件得:对任意e0 存在N使 当nN时有:|(An+1-An)(Bn+1-Bn)-L|e,即|(An+1-An)(Bn+1-Bn)|e;又Bn0递增且有n--+∞时Bn--+∞,原式化为:|An+1-An|e*(Bn+1-Bn)......⑴;固定e,则存在N1=N,当nN1时,有-e*BN+|AN|e*Bn即|AN|e*(BN+Bn) ..........⑵重要|An|=|An-An-1|+|An-1-An-2|+....+|AN+1-AN|+|AN|,代入⑴式,得:=e*(Bn-Bn-1)+.....+e(BN+1-BN)+|AN|,代入⑵式,得:e*(Bn-BN)+e*(Bn+BN)即|An|2e*Bn故|(An)(Bn)-0|2e由数列定义知lim(An)(Bn)=02)当L=C (C!=0)时即有lim(An+1-An)(Bn+1-Bn)=C,令Cn=An-C*Bn,显然有lim(Cn+1-Cn)(Bn+1-Bn)=0,由1)得:故lim(Cn)(Bn)=0,即有lim(An)(Bn)=C,3)当L=+∞(L=-∞时类证)时存在N,当nN时有(An+1-An)(Bn+1-Bn)1得出AnBn0,且满足An0递增且有n--+∞时An--+∞所以lim(Bn+1-Bn)(An+1-An)=0+ (0+即从正数趋近于0)由1)得:lim(Bn)(An)=0+故lim(An)(Bn)=+∞证毕