June 4th, 2019, 06:29 AM  #1 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,821 Thanks: 643 Math Focus: Yet to find out.  asymptotics
If $$ \lim\limits_{n \to \infty} \dfrac{f(n)}{g(n)} = \dfrac{a}{b}$$ with $a \ne b$, $b \ne 0$ then $$b f(n) \sim a g(n) $$ ?....

June 4th, 2019, 07:16 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 534 Thanks: 81 
If you set $\displaystyle a/b$ inside the limit then it is done .

June 4th, 2019, 07:38 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,821 Thanks: 643 Math Focus: Yet to find out. 
no but then why define $\sim$ the way it is

June 4th, 2019, 08:09 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2642 Math Focus: Mainly analysis and algebra  
June 6th, 2019, 11:02 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,821 Thanks: 643 Math Focus: Yet to find out. 
No I get it's a stupid question but I meant $a \ne 0$ also. Basically I have a function with known limit which is not unity and I want to infer how it grows without having to juggle terms in the limit

June 12th, 2019, 12:47 AM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,821 Thanks: 643 Math Focus: Yet to find out. 
The point is that the limit either a) converges to a constant greater than 0, diverges to $\pm \infty$ or converges to 0. If a) then $f \sim g$, b) $f$ grows faster, c) $g$ grows faster.

June 12th, 2019, 05:21 AM  #7  
Math Team Joined: Jul 2011 From: Texas Posts: 2,940 Thanks: 1545  Quote:
$\displaystyle \lim_{n \to \infty} \dfrac{b \cdot f(n)}{a \cdot g(n)}= 1$  
June 12th, 2019, 05:40 AM  #8 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,821 Thanks: 643 Math Focus: Yet to find out.  Yes.. just in the asymptotic sense, the constants $a,b$ don't factor into the expression $f \sim g$ because they're negligible in the limit.


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