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Is it possible to evaluate the limit $$ \lim\limits_{\epsilon \to + 0} \epsilon^2\int\limits_0^\epsilon \frac{r^{d-1} \, dr}{(r^2+\epsilon^2)^{\frac{d+2}{2}}}, \quad d > 2, $$ without evaluating the integral for each $\epsilon > 0$? Estimating the numerator in the integral by $r(r^2+\epsilon^2)^{\frac{d-2}{2}}$ I can obtain an upper estimate $1/4$. The L'Hopital rule doesn't help.

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I would start with $$\lim\limits_{\epsilon \to + 0} \epsilon^2\int\limits_0^\epsilon \frac{r^{d-1} \, dr}{(r^2+\epsilon^2)^{\frac{d+2}{2}}} =\lim\limits_{\epsilon \to + 0} \int\limits_0^1 \frac{u^{d-1} \, du}{(u^2+1)^{\frac{d+2}{2}}}.$$ Then, clearly, we need to evaluate the last expression, which is not that evident.

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