Could someone show me how to solve the following limits algebraically:
For x > 1, lim as n approaches infinity of
a) [(1 + x^(1/n))/2]^(2n) = x
b) [(1 + x^(1/n))/2]^(n) = sqrt(x)
They seem very similar to the sequence (1 + (a/n))^n whose limit is e^a, but I don't see how to manipulate it accordingly.
If you could show me how to solve these, I would really appreciate it.
Thanks in advance.
That's what the limits equal. Sorry about the confusion.
I need the limit of [(1 + x^(1/n))/2]^(2n), which equals x, and the limit of [(1 + x^(1/n))/2]^(n), which equals the square root of x.
I just don't know how to show them with algebra.
PerfectHacker, thanks for the response, but I don't really see what you're doing. Could you elaborate? Basically, how does it simplify to x?