Limit as x approaches infinity.

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

10logx

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

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

Ray Vickson
Homework Helper
Dearly Missed

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

10logx

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%.

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