Limit as x approaches infinity.

E.g., there is a thread here on PF where ##\log## is used with base 2 (for a "binary entropy function" or something like that).The "base" of a logarithm is a bit of a technicality. It is really just a scaling factor: if ##a>0## and ##ae1##, then##\log_a(x)=k\log_b(x)##, with ##k=\log_a(b)##. So if you know the logarithm to one base, then you can get the logarithm to another base just by multiplying by a constant factor. And if you know the logarithm to a base ##b##, then you can get the logarithm to base
  • #1
SPhy
25
0

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)?
 
Last edited:
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  • #2
I would try rescaling the independent variable as ##x=e^{\xi}##.
 
  • #3
SPhy said:

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%.
 
Last edited:

Related to Limit as x approaches infinity.

What is the definition of a limit as x approaches infinity?

The limit as x approaches infinity is the value that a function approaches as the input x gets closer and closer to infinity. It is denoted by the symbol "lim" and is used to describe the behavior of a function at infinity.

How do you evaluate a limit as x approaches infinity?

To evaluate a limit as x approaches infinity, you can either use algebraic techniques such as factoring and simplifying, or you can use the rules of limits which state that the limit of a sum, difference, product, or quotient is equal to the sum, difference, product, or quotient of the individual limits.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches infinity from one direction (either positive or negative), while a two-sided limit considers the behavior of a function as the input approaches infinity from both directions.

What types of functions have a limit as x approaches infinity?

Most continuous functions have a limit as x approaches infinity, meaning that the function approaches a specific value as x gets larger and larger. However, some functions, such as oscillating or piecewise-defined functions, may not have a limit at infinity.

How does the concept of a limit as x approaches infinity relate to real-world applications?

The concept of a limit as x approaches infinity is used in various fields of science and engineering, such as physics, chemistry, and economics. It helps in predicting the behavior of systems that involve continuously changing variables, such as population growth, radioactive decay, and fluid flow.

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