0/0 type restriction
Using L'H?pital's law, the upper and lower derivatives are derived respectively.
Original limit = lim (x->; 0)( 1/( 1+x))/ 1 = 1
ln[( 1+x)/x]= ln( 1+x)-lnx
When x approaches 0, the limit can be found separately.
= the result of negative infinity
Correspondence of n
Generally speaking, n increases with the decrease of ε, so n is often written as N(ε) to emphasize the dependence of n on ε change. But this does not mean that n > is exclusively determined by ε: (for example, if n >; N makes | xn-a | < ε hold, then obviously n & gtN+ 1, n >2N and so on also make | xn-a |.