x趨近于0,lim[(e^tanx-e^x)/(x^k)]
=lim{e^x*[(e^(tanx-x)-1]/x^k}
=lim [e^(tanx- x)/x^k]
=lim(tanx- x)/x^k
=lim{[sec(x)]^2-1}/(kx^(k-1))
=lim(tanx)^2/(kx^(k-1))
=lim [x^2/(kx^(k-1))]
2=k-1,k = 3
寫得比較詳細(xì),具體做時(shí)可以簡(jiǎn)寫
若熟悉x趨近于0,tanx- x 等價(jià)于 (1/3)x^3 (泰勒級(jí)數(shù)展開）
=lim(tanx- x)/x^k = (1/3)limx^(3-k)=常數(shù) ,k=3