Question

求助高数学霸！！！

Answer 1

lim [(3+x)/(6+x)]^[(x-1)/2]
=lim e^{ ln [(3+x)/(6+x)]^[(x-1)/2] }
=e^{lim ln [(3+x)/(6+x)]^[(x-1)/2] }
考虑
lim ln [(3+x)/(6+x)]^[(x-1)/2]
=lim (x-1)/2 * [ln(3+x)-ln(6+x)]
=lim [ln(3+x)-ln(6+x)] / [2/(x-1)]
明显极限为0/0型，根据L'Hospital法则
=lim [ln(3+x)-ln(6+x)]' / [2/(x-1)]'
=lim [1/(3+x)-1/(6+x)] / [2/(x-1)^2]
=lim [3/(3+x)(6+x)] / [2/(x-1)^2]
=(3/2) * lim (x-1)^2 / (x+3)(x+6)
=3/2
于是，原极限=e^(3/2)
有不懂欢迎追问