# How find this limit: $\displaystyle \lim_{n\to\infty} \int_0^1 (1+x)^{-n-1}e^{x^2}\ dx$

I wish to find: $$\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( > n=1,2,\cdots)$$

I maybe have to evaluate: $$\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx$$

But I can't, can someone give me some help?

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We can partially evaluate the integral, by integrating by parts,

\begin{align} \int_0^1 \frac{n}{(1+x)^{n+1}}e^{x^2}\,dx &= \left[-\frac{e^{x^2}}{(1+x)^n} \right]_0^1 + \int_0^1 \frac{2xe^{x^2}}{(1+x)^n}\,dx\\ &= 1 - \frac{e}{2^n} + \int_0^1 \frac{2xe^{x^2}}{(1+x)^n}\,dx \end{align}

The last two terms converge to $0$, so the limit is $1$.

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Let $u=nx$ then we have $$n\int_0^1 (1+x)^{-(n+1)}e^{x^2}dx=\int_0^n(1+u/n)^{-(1+n)}e^{(u/n)^2}du\to\int_0^\infty e^{-u}du=1$$

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