Calculate limit value with several variables

[GravityX]

Homework Statement

Calculate the limit of $P$ when $a_0 \rightarrow 0$ and $M,n \rightarrow \infty$ with $a=a_0n$ and $L=a_0*M$.

Relevant Equations

none

Hi,

I had to calculate the entropy in a task of a lattice gas and derive a formula for the pressure from it and got the following

$$P=\frac{k_b T}{a_0}\Bigl[ \ln(\frac{L}{a_0}-N(n-1)-\ln(\frac{L}{a_0}-nN) \Bigr]$$

But now I am supposed to calculate the following limit

$$\lim\limits_{a_0 \rightarrow \infty}{} \lim\limits_{M \rightarrow \infty}{} \lim\limits_{n \rightarrow \infty}{\frac{k_b T}{a_0}\Bigl[ \ln(\frac{L}{a_0}-N(n-1)-\ln(\frac{L}{a_0}-nN) \Bigr]}$$

So not the limit for $a_0$ , $M$ and $n$ but all at the same time.

Should I first calculate the limit for one, say for $a_0$ and what I got for that, the limit for $M$ or better said $L$ etc?

 

[andrewkirk]

If the simultaneous limit exists, it doesn't matter what order you take the limits in. The eventual answer must be the same. Although some orders may be easier than others.

Where they exist, first calculate limits for components of the formula, and replace those components by their limits in the formula. That's generally valid as long as both the overall limit and the component's limit exist.

So for instance, $\lim_{a_0\to\infty} \frac L{a_0}$ is easy.
Another hint, for the expression in square brackets, use the fact that $\log a - \log b = \log\left(\frac ab\right)$ and then rewrite the fractional expression you're taking the log of as $1 + \frac{1}{denominator}$. You'll find it easier to take limits that way.

By the way, there is no $M$ in your formulas. I presume you mean $N$.

 

[BvU]

By the way, there is no $M$ in your formulas. I presume you mean $N$.

$a=a_0n$ and $L=a_0*M$.

Messy. First I see $a_0\downarrow 0$, then $a_0\uparrow \infty$. Typos ?

 

[SammyS]

Messy. First I see $a_0\downarrow 0$, then $a_0\uparrow \infty$. Typos ?

Also:
You have unbalanced parentheses.