Natural log limits as n approaches infinity

In summary, the limit of ertain function x as n approaches infinity is x_n = \frac{{\ln (1 + {n^{\frac{1}{2}}} + {n^{\frac{1}{3}}})}}{{\ln (1 + {n^{\frac{1}{3}}} + {n^{\frac{1}{4}}})}} = \frac{{\ln \left( {{n^{\frac{1}{2}}} \cdot (\frac{1}{{{n^{\frac{1}{2}}}}} + 1 + \frac{1}{{{n^{\frac{1}{6}}}}})} \right)}}{{\ln \left( {{
  • #1
MSG100
43
0
My question is:

Show the limit of
[tex]x_{n}=\frac{ln(1+\sqrt{n}+\sqrt[3]{n})}{ln(1+\sqrt[3]{n}+\sqrt[4]{n})}[/tex]
as n approaches infinity


Solution:
[tex]{x_n} = \frac{{\ln (1 + {n^{\frac{1}{2}}} + {n^{\frac{1}{3}}})}}{{\ln (1 + {n^{\frac{1}{3}}} + {n^{\frac{1}{4}}})}} = \frac{{\ln \left( {{n^{\frac{1}{2}}} \cdot (\frac{1}{{{n^{\frac{1}{2}}}}} + 1 + \frac{1}{{{n^{\frac{1}{6}}}}})} \right)}}{{\ln \left( {{n^{\frac{1}{3}}} \cdot (\frac{1}{{{n^{\frac{1}{3}}}}} + 1 + \frac{1}{{{n^{\frac{1}{{12}}}}}})} \right)}} = [/tex]


[tex]= \frac{{\ln {n^{\frac{1}{2}}} + \ln (\frac{1}{{{n^{\frac{1}{2}}}}} + 1 + \frac{1}{{{n^{\frac{1}{6}}}}})}}{{\ln {n^{\frac{1}{3}}} + \ln (\frac{1}{{{n^{\frac{1}{3}}}}} + 1 + \frac{1}{{{n^{^{\frac{1}{{12}}}}}}})}} = \frac{{\frac{1}{2}\ln n + \ln (\frac{1}{{{n^{\frac{1}{2}}}}} + 1 + \frac{1}{{{n^{\frac{1}{6}}}}})}}{{\frac{1}{3}\ln n + \ln (\frac{1}{{{n^{\frac{1}{3}}}}} + 1 + \frac{1}{{{n^{^{\frac{1}{{12}}}}}}})}} =[/tex]

[tex]= \frac{{\frac{1}{2} + \frac{1}{{\ln n}} \cdot \ln (1 + \frac{1}{{{n^{\frac{1}{2}}}}} + \frac{1}{{{n^{\frac{1}{6}}}}})}}{{\frac{1}{3} + \frac{1}{{\ln n}} \cdot \ln (1 + \frac{1}{{{n^{\frac{1}{3}}}}} + \frac{1}{{{n^{\frac{1}{{12}}}}}})}}[/tex]


Now I'm stuck.
Is this the right way to do it?
 

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  • #2
That looks good so far... as n goes to infinity, what happens to 1/ln(n), and what happens to ln(1+1/n1/2+1/n1/6) and the similar term in the denominator?
 
  • #3
How did you (OP) go from (1/2)ln(n) to 1/2 + 1/ln(n) ? And same question for the similar part of the denominator?
 
  • #4
I would continue from here:
[tex]= \frac{{ \frac 1 2 \ln n + \ln (\frac{1}{{{n^{\frac{1}{2}}}}} + 1 + \frac{1}{{{n^{\frac{1}{6}}}}})}}{{ \frac 1 3 \ln n + \ln (\frac{1}{{{n^{\frac{1}{3}}}}} + 1 + \frac{1}{{{n^{^{\frac{1}{{12}}}}}}})}} [/tex]
by factoring ln(n) from each term in the numerator and each term in the denominator, with the result looking like this:
$$ \frac{\frac 1 2 ln(n)(1 + \text{other stuff})}{\frac 1 3 ln(n)(1 + \text{some other stuff})}$$
As long as you can convince yourself the <other stuff> goes to zero as n gets large, you should be able to get the desired answer of 3/2.
 
  • #5
Mark he just divided the numerator and denominator by ln(n)
 
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  • #6
OK, I didn't see that. It looked to me like he was using an invalid log property. What he did is pretty close to what I'm suggesting in post #4.
 
  • #7
I liked your first 3 steps until you got to ##\frac{(1/2)logn + stuff}{ 1/3logn + stuff}##. What I'm calling "stuff" will all go to 0 as n ## \rightarrow \infty##.

So from that point you can go directly to the answer of 3/2 -- I don't thing you need to fuss any further.

You should probably explain why the stuff goes to 0.
 
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  • #8
Many thanks to all of you for the quick response! I can see it now!

These kind of problems are quite new for me so maybe I overdo it.

I need to do some more of these so I can get some grip of it.
 

1. What is a natural log limit as n approaches infinity?

A natural log limit as n approaches infinity is a mathematical concept that involves taking the natural logarithm of a function as the input variable approaches infinity. This can be written as lim ln(n) as n approaches infinity.

2. What is the purpose of finding a natural log limit as n approaches infinity?

The purpose of finding a natural log limit as n approaches infinity is to understand the behavior of a function as the input variable becomes larger and larger. This can help in analyzing the long-term trend or growth of a function.

3. How is a natural log limit as n approaches infinity evaluated?

A natural log limit as n approaches infinity is evaluated by taking the limit of ln(n) as n approaches infinity. This can be done by simplifying the function, using L'Hopital's rule, or using other methods such as substitution or factoring.

4. What are the common applications of natural log limits as n approaches infinity?

Natural log limits as n approaches infinity are commonly used in various fields such as mathematics, physics, finance, and engineering. They can be used to model growth rates, analyze the behavior of complex systems, and make predictions about future trends.

5. Are there any limitations to natural log limits as n approaches infinity?

One limitation of natural log limits as n approaches infinity is that they can only be applied to functions that have a natural logarithm. Also, these limits may not always provide an accurate representation of the behavior of a function in the long-term, as they only consider the behavior as the input variable becomes infinitely large.

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