Limit as x approaches infinity.

[SPhy]

Homework Statement

Compute Limit as x--> infinity of (logx)^(log(logx)) / x

The Attempt at a Solution

Graphically, I see that the answer is perhaps zero, but I am not sure how to approach this algebraically. I worked at this for a couple hours, trying L'Hospital's rule but that did not really lead me anywhere. Is there some tricky substitution I am missing?

Can I do this?

Rewrite (logx)^log(log(x))
as

10^(log(logx))2

and then x as

10^logx

and evaluate the limit of the exponents? (as in as a ratio)?

 

[NFuller]

I would try rescaling the independent variable as $x=e^{\xi}$.

 

[Ray Vickson]

Homework Statement

Compute Limit as x--> infinity of (logx)^(log(logx)) / x

The Attempt at a Solution

Graphically, I see that the answer is perhaps zero, but I am not sure how to approach this algebraically. I worked at this for a couple hours, trying L'Hospital's rule but that did not really lead me anywhere. Is there some tricky substitution I am missing?

Can I do this?

Rewrite (logx)^log(log(x))
as

10^(log(logx))2

and then x as

10^logx

and evaluate the limit of the exponents? (as in as a ratio)?

Isn't $\log(x)$ the natural logarithm $\ln (x)$? Nowadays that is usually the case, with $\log$ and $\ln$ just being two alternative notations for the same function. Then, if alternative bases are needed they are indicated explicitly, such as $\log[10](x)$ or $\log_{10}(x).$

However, if you are looking at a 50-100 year-old book that might not be the case, and even today there may be some exceptions: "usual" does not mean 100%.