Calculate limit value with several variables

In summary: Also:In summary, you are calculating the entropy and pressure of a lattice gas, but you are also supposed to calculate a limit. You first calculate the limits for the entropy and pressure, and then you replace the limits for the entropy and pressure with the limit for the whole thing.
  • #1
GravityX
19
1
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?
 
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  • #2
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##.
 
  • #3
andrewkirk said:
By the way, there is no ##M## in your formulas. I presume you mean ##N##.
GravityX said:
##a=a_0n## and ##L=a_0*M##.
Messy. First I see ##a_0\downarrow 0##, then ##a_0\uparrow \infty##. Typos ?
 
  • #4
BvU said:
Messy. First I see ##a_0\downarrow 0##, then ##a_0\uparrow \infty##. Typos ?
Also:
You have unbalanced parentheses.
 

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