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If it’s possible, I want to write this limit based on delta function

$$ \lim_{t\rightarrow 0}\frac{e^{-u/t}}{t^2},\qquad u>0 $$

would you mind helping me?

Michael Hardy
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user166453
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  • Can you clarify on what it means to "write this limit based on delta function"? You means the Dirac delta function in term of $u$? But then $u>0$. Also, what's your thought on the problem so far? – Gina Jul 26 '14 at 17:39
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    The obvious thing to try is to see how the the expression behaves under an integral sign. – David H Jul 26 '14 at 17:59

1 Answers1

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Hint:

The Dirac delta function has the limit representation,

$$\delta{(x)}=\lim_{\epsilon\to0}\frac{1}{2\epsilon}e^{-\frac{|x|}{\epsilon}}.$$

Then, for some fixed but arbitrary function $f(x)$, we may represent its inner product with $\delta{(x)}$ by the limit:

$$\int_{-\infty}^{\infty}\mathrm{d}x\,\delta(x)f(x)=\lim_{\epsilon\to0}\int_{-\infty}^{\infty}\mathrm{d}x\,\frac{1}{2\epsilon}e^{-\frac{|x|}{\epsilon}}\,f(x)\,.$$

Integrate by parts and see if you recognize anything.

David H
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