Proving Limit of ln(x)/x as x-->∞ is 0

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In summary, the conversation discusses how to prove that the limit of (ln x/x) as x approaches to infinity equals 0 using the definition of ln x as an integral. The participants consider using L'Hopital's rule and a rigorous proof with epsilon and delta, ultimately concluding that L'Hopital's rule is the best approach.
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battousai
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Homework Statement



Prove that the limit of (ln x/x) as x approaches to infinity equals 0 starting with the definition of ln x as an integral. (How do you type a limit with LaTex?)

Homework Equations




[tex]\int \frac{1}{x}\,dx[/tex] = ln lxl + C

The Attempt at a Solution



I don't even know how to begin the solution. Unfortunately we don't really do much proofs in math Calc BC class. I found this problem online and I thought it was interesting. Any tips to get me started would be appreciated.
 
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  • #2
Do you mean a rigorous proof ?

Anyway divide the integral by x and use L'hospitals.
 
  • #3
Hmm I would think a rigorous proof. I did think of L'Hopital's, but I thought that was too easy.

It's just a problem I saw online so IDK exactly what kind of proof it's looking for.
 
  • #4
You mean with epsilons and delta's ?

Well using the precise definition of a limit to do the proof is not too easy.
Write down your question precisely with epsilon and delta then see if you could make some approximations. You are likely going to run into some difficulties.

I think L'hostipals is the way to go.
 
  • #5
[tex]\frac{\int\frac{1}{x}dx}{x}[/tex]
Now use L'Hopital Rule..
You will be [tex]\frac{1}{x}[/tex]
Then you allow ur limit to tend to infinity.
 

1. What is the limit of ln(x)/x as x approaches infinity?

The limit of ln(x)/x as x approaches infinity is 0. This means that as x gets larger and larger, the value of ln(x)/x gets closer and closer to 0.

2. How do you prove that the limit of ln(x)/x as x approaches infinity is 0?

To prove that the limit of ln(x)/x as x approaches infinity is 0, we can use the definition of a limit. This involves showing that for any small positive number, there exists a corresponding value of x such that ln(x)/x is less than that number. We can also use algebraic manipulations and the properties of logarithms to simplify the expression and show that it approaches 0 as x gets larger.

3. Can the limit of ln(x)/x as x approaches infinity be calculated without using the definition of a limit?

Yes, there are other methods to calculate the limit of ln(x)/x as x approaches infinity. One method is to use L'Hospital's rule, which states that the limit of the quotient of two functions can be found by taking the derivative of the numerator and denominator and then evaluating the limit. In this case, taking the derivative of ln(x) and x will result in 1/x, which approaches 0 as x approaches infinity.

4. How does the graph of ln(x)/x change as x approaches infinity?

As x approaches infinity, the graph of ln(x)/x approaches the x-axis. This means that the graph gets closer and closer to the x-axis but will never touch it. The graph is also concave down, meaning that it curves downwards as x gets larger.

5. What are the practical applications of proving the limit of ln(x)/x as x approaches infinity is 0?

The limit of ln(x)/x as x approaches infinity has various applications in mathematics, physics, and engineering. It is used in the analysis of algorithms and the study of asymptotic complexity. It is also used in the study of exponential growth and decay, as well as in the calculation of derivatives and integrals involving logarithmic functions. Additionally, it has applications in the study of heat transfer, fluid dynamics, and electrical circuits.

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