Please help me in solving the limits question. Now since it is infinity -  infinity form so I took Lcm then I didn't understand what to do

2081 Points
4 years ago
first note that the first term in question is 1/[h*(8+h)^(1/3)] and not 1/[h^3*(8+h)^(1/2)] as it initially seemed to me.
now, put h=8x.
limit becomes
(1/16)*lim x ->0 [(1+x)^(-1/3) – 1]/x
now put 1+x=y
so we have
(1/16)*lim y ->1 [(y)^(-1/3) – 1]/(y – 1)
by the first result of limits, we know that lim y ->1 [(y)^(-1/3) – 1]/(y – 1)= na^(n-1), where n= – 1/3 and a=1.
so, limit= (1/16)*(-1/3)
= – 1/48