Evaluate $\exp{(a/T)} / T$ for $T \rightarrow 0$

Question

How do I evaluate the expression:

$\exp{(a/T)} / T$ for $T \rightarrow 0$

If I use L'Hôpital's rule I just get:

$\frac{\frac{d}{dT}\exp{(a/T)}}{\frac{d}{dT}T} = \frac{-a\exp{a/T}}{T^2}$

I know I am suppose to get zero. But Im not really sure how to get it..

Well it is actually an energi divided by boltzmanns konstant. But now you are mentioning it. I guess I can use T<<a. — Nillo, Oct 22 '15 at 13:18

score 2 · Accepted Answer · answered Oct 22 '15 at 13:21

Setting $\frac{1}{T} = x$, we're interested in the limit

$$ \lim_{x \to \infty} x e^{ax}$$

It's a simple exercise to show that this goes to $0$ if $a < 0$, and infinity if $a \ge 0$. The latter is obvious. For the former,

\begin{align} \lim_{t \to 0} \frac{e^{a/t}}{t} &= \lim_{t \to 0} \frac{1/t}{e^{-a/t}} \\ &= \lim_{t \to \infty}\frac{-1/t^2}{a e^{-a/t}/t^2}\\ &= \lim_{t \to \infty} \frac{-e^{a/t}}{a}\\ &= 0 \end{align}

Where the last equality is due to $a < 0$ and the ones before that are l'Hopital's rule.

Evaluate $\exp{(a/T)} / T$ for $T \rightarrow 0$

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