Calculate limit value with several variables

AI Thread Summary
The discussion revolves around calculating a complex limit involving multiple variables related to entropy in a lattice gas. The formula for pressure is provided, and the user seeks guidance on whether to calculate the limits for each variable sequentially or simultaneously. It is suggested that calculating limits for components of the formula first can simplify the process, as long as both the overall limit and the individual limits exist. Additionally, there is a clarification regarding a potential typo in the variables used, specifically the mention of "M" instead of "N." The conversation highlights the importance of careful notation and the potential for confusion in mathematical expressions.
GravityX
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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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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##.
 
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 ?
 
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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