Finding Limits of Functions with Multiple Sets of Variables

[Mathoholic!]

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)=\frac{n}{q}

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

 

[JonDrew]

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) / (q-n)^2

because then you would have multiple variables locked upstairs and downstairs and it would become a mess with partial derivations.

 

[rollingstein]

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

 

[Curious3141]

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)=\frac{n}{q}

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

You have to define your n and q better.

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.

 

[pwsnafu]

How do I avaluate the following limit (if possible):

lim f(n,q) as n→∞ and q→∞

Do you mean $\lim_{n \rightarrow \infty}\lim_{q \rightarrow \infty} \frac{n}{q}$ or $\lim_{q \rightarrow \infty} \lim_{n \rightarrow \infty} \frac{n}{q}$?

I was struck by a problem: What if the variables of the function belong to different sets of numbers?

The general theory is that of nets.

 

[JonDrew]

You have to define your n and q better.

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.

Very cool! Thanks for the correction.