为了说明方便,设 t = x - ln2。那么,当 x →+∞ 时,t →+∞
原极限
= lim[(t + 2ln2)/t]^(t+ln2)
=lim(1 + 2ln2/t)^t * (1 + 2ln2/t)^(ln2)
=lim(1 + 2ln2/t)^[t/(2ln2) * (2ln2)] * lim(1+2ln2/t)^(ln2)
=lim(1 + 2ln2/t)^[t/(2ln2) * (2ln2)] * lim(1+0)^(ln2)
=lim(1 + 2ln2/t)^[t/(2ln2) * (2ln2)]
再设 t/(2ln2) = y,那么当 t →+∞ 时,y = lim[t/(2ln2)] → +∞。则上面的极限:
=lim(1 + 1/y)^(y * 2ln2)
=lim[(1+1/y)^y]^(2ln2)
=e^(2ln2)
=e^ln4
=4
图中的写法正确啊,具体参考下图