How to Prove the Limit of a Logarithmic Function Ratio?

In summary, a limit in mathematics is a value that a function or sequence approaches as the input or index approaches a certain value. It is used to describe the behavior of a function or sequence near a specific point or at infinity. A limit can be used to find a function by evaluating the behavior of the function at a specific point and determining the corresponding limit. The process of finding a function from a limit involves evaluating the limit of a given function at a specific point or at infinity. Not all limits can be used to define a function, and some may have multiple possible functions. Finding a function from a limit can be useful in real-world applications, such as predicting and modeling the behavior of systems and processes.
  • #1
jillna
2
0
Posted: Tue, 20 Jul 2010 14:49:50 Post subject: find a function from the limit
Suppose:

[late]\lim_{x \rightarrow \infty} \frac{log f_1(x)}{ log x } = c_1[/late]

and

[late]\lim_{x \rightarrow \infty} \frac{log f_2(x)}{ log x } = c_2[/late]

if [late]c_2>c_1[/late] then

[late]\lim_{x \rightarrow \infty} \frac{ log f_1(x) + f_2(x) ] }{ log x } = c_2 [/late]

can anyone tell me how to prove this rigorously?

tks
 
Last edited:
Physics news on Phys.org
  • #2
The general idea involves using the fact that fi(x) ~xci. Therefore f1(x)/f2(x) -> 0 as x ->∞. As a result the numerator in your last expression ends up behaving like logf2(x).

Good luck.
 

FAQ: How to Prove the Limit of a Logarithmic Function Ratio?

What is a limit in mathematics?

A limit in mathematics is a value that a function or sequence approaches as the input or index approaches a certain value. It is used to describe the behavior of a function or sequence near a specific point or at infinity.

How is a limit used to find a function?

A limit can be used to find a function by evaluating the behavior of the function at a specific point and determining the corresponding limit. This limit can then be used to define the function at that point and extend it to a larger domain.

What is the process of finding a function from a limit?

The process of finding a function from a limit involves evaluating the limit of a given function at a specific point or at infinity. This limit is then used to define the function at that point, and the function is extended to a larger domain by considering its behavior near the defined point.

Can a function always be found from a given limit?

No, not all limits can be used to define a function. Some limits may not have a corresponding function, while others may have multiple possible functions depending on the behavior of the function near the defined point.

How can finding a function from a limit be useful in real-world applications?

Finding a function from a limit can be useful in real-world applications such as physics, engineering, and economics. It can be used to model and predict the behavior of systems and processes, and can also help in solving problems involving rates of change and optimization.

Similar threads

I Proving linear independence of two functions in a vector space
Replies
5
Views
2K
Prove that ## li(x)\sim \frac{x}{\log x} ##.
Replies
3
Views
1K
B Understanding exponentiation and logarithms together
Replies
12
Views
2K
B I don't recognize this limit of Riemann sum
Replies
2
Views
2K
A Proving that this integral is divergent
Replies
2
Views
811
Limit problem involving two circles and a line
Replies
8
Views
1K
Find limit of multi variable function
Replies
10
Views
1K
I Bounding modulus of complex logarithm times complex power function
Replies
4
Views
1K
I Calculate limits as distributions
Replies
4
Views
2K
Verify the convergence or divergence of a power series
Replies
17
Views
2K

Hot Threads

Recent Insights

Back
Top