which grows faster? f(x) = (ln(x))^(1/2) or g(x) = ln(x^(1/2)) as x approaches infinity?

Question

which grows faster? f(x) = (ln(x))^(1/2) or g(x) = ln(x^(1/2)) as x approaches infinity?

Answer 1

If you have covered derivatives, this is straightforward.

We just need to take the derivatives of f and g and figure out which one, in the long run, is larger.

d/dx (√ln(x)) = )(1/2lnx)1/x

= 1/ (2x ln(x))

d/dx ln(√x) = 1/√x * 1/2√x = 1/2x

Since 2xln(x)>2x in the long run, 1/2xln(x) < 1/2x in the long run, so the second function grows faster in the long run (ln(√x) = ln(x^1/2).