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Finding a limitby Mathoholic!
Tags: limit 
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#1
Jan2713, 01:13 AM

P: 49

I'm familiarized with finding limits of most kinds of functions. I was struck by a problem: What if the variables of the function belong to different sets of numbers?
My point being, given the function: f(n,q)=[itex]\frac{n}{q}[/itex] With n belonging to the set of natural numbers and q belonging to the set of rational numbers. How do I avaluate the following limit (if possible): lim f(n,q) as n→∞ and q→∞ This may be a silly question but care to answer please. Thank you 


#2
Jan2713, 01:56 AM

P: 83

I am not 100% certain but I believe L'Hospital's rule still applies to multivariable functions if both the top and bottom functions are going to infinity. That would make:
lim f(n,q) as n→∞ and q→∞ = 1 only because you can take the derivative of what is upstairs and the derivative of what is downstairs with simplicity. I don't think this method would work if you had say, a function like: f(n,q) = (n+q) / (qn)^2 because then you would have multiple variables locked upstairs and downstairs and it would become a mess with partial derivations. 


#3
Jan2713, 02:05 AM

PF Gold
P: 330

That function is discontinous along one axis. Don't think L'Hospital can be used then.



#4
Jan2713, 02:06 AM

HW Helper
P: 2,943

Finding a limit
As an example, let's take the limit ##\lim_{x \rightarrow \infty} \frac{\lfloor x \rfloor}{x}##, where ##x \in \mathbb{R}## You can't simply apply L' Hopital's because the numerator is a discontinuous function. But you can find the limit by defining the fractional part of x as ##\{x\}## and rewriting the limit as: ##\lim_{x \rightarrow \infty} \frac{x  \{x\}}{x} = 1  \lim_{x \rightarrow \infty} \frac{\{x\}}{x} = 1## So a lot depends on exactly what you intend the numerator and denominator to signify. 


#5
Jan2713, 02:09 AM

Sci Advisor
P: 820




#6
Jan2713, 12:28 PM

P: 83




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