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 June 4th, 2019, 07:29 AM #1 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,847 Thanks: 661 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, 08:16 AM #2 Senior Member   Joined: Dec 2015 From: Earth Posts: 817 Thanks: 113 Math Focus: Elementary Math If you set $\displaystyle a/b$ inside the limit then it is done . June 4th, 2019, 08:38 AM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,847 Thanks: 661 Math Focus: Yet to find out. no but then why define $\sim$ the way it is June 4th, 2019, 09:09 AM   #4
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 Originally Posted by Joppy If $$\lim\limits_{n \to \infty} \dfrac{f(n)}{g(n)} = \dfrac{a}{b}$$ with $a \ne b$, $b \ne 0$ then $$b f(n) \sim a g(n)$$ ?....
$a=0$, $f(x)=x$, $g(x)=x^2$ June 7th, 2019, 12:02 AM #5 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,847 Thanks: 661 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, 01:47 AM #6 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,847 Thanks: 661 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, 06:21 AM   #7
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Quote:
 Originally Posted by Joppy If $$\lim\limits_{n \to \infty} \dfrac{f(n)}{g(n)} = \dfrac{a}{b}$$ with $a \ne b$, $b \ne 0$ then $$b f(n) \sim a g(n)$$ ?....
$\displaystyle \dfrac{b}{a} \lim_{n \to \infty} \dfrac{f(n)}{g(n)}= \dfrac{b}{a} \cdot \dfrac{a}{b}$

$\displaystyle \lim_{n \to \infty} \dfrac{b \cdot f(n)}{a \cdot g(n)}= 1$ June 12th, 2019, 06:40 AM   #8
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 Originally Posted by skeeter $\displaystyle \dfrac{b}{a} \lim_{n \to \infty} \dfrac{f(n)}{g(n)}= \dfrac{b}{a} \cdot \dfrac{a}{b}$ $\displaystyle \lim_{n \to \infty} \dfrac{b \cdot f(n)}{a \cdot g(n)}= 1$
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. Tags asymptotics Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post kikou Applied Math 0 March 30th, 2012 07:30 PM alex2010 Real Analysis 1 October 19th, 2010 09:38 AM

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