lim(x->∞) [(x+2)/(x+3)]^x=?
记:J=[(x+2)/(x+3)]^x
lnJ = x ln[(x+2)/(x+3)] = x ln[(1+2/x)/(1+3/x)]
= [ln(1+2/x) - ln(1+3/x)]/(1/x)
= [ln(1+2/x)]/(1/x) - [ln(1+3/x)]/(1/x) //:当:x->∞ 变成 0/0 型的不定式,采用洛必达法则:
lim(x->∞) lnJ = lim(x->∞) [-2/x^2/{(1+2/x)(-1/x^2)}] - [-3/x^2/{(1+3/x)(-1/x^2)}]
= lim(x->∞) [2/(1+2/x) - 3/(1+3/x)]
= 2-3=-1
得到:lim(x->∞) lnJ = -1
因此:lim(x->∞) J = 1/e
即:lim(x->∞) [(x+2)/(x+3)]^x = 1/e .