How to Prove the Limit of a Logarithmic Function Ratio?

[jillna]

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

 

[mathman]

The general idea involves using the fact that f_i(x) ~x^c_i. 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.