Limit of an Integral

[snipez90]

1. The problem statement, all variables and given/known data
Consider the function f: R -> R, f(x) = (x^2 + 1)e^x . Find the limit \mathop {\lim }\limits_{n\rightarrow\infty}n\int_{0}^{1}\left(f\ left(\frac{x^2}{n}\right) - 1\right).


2. Relevant equations
e^x > x + 1 for nonzero real x

3. The attempt at a solution
After a bit of algebra, we find that the original limit is
\mathop {\lim }\limits_{n\rightarrow\infty}\frac{1}{n}\int_{0}^{ 1}e^{\frac{x^2}{n}}x^4\,dx + \mathop {\lim }\limits_{n\rightarrow\infty}n\int_{0}^{1}\left(e^ {\frac{x^2}{n}}-1\right)\,dx.

In the first limit, the integrand is increasing on [0,1], so we have
0 \leq \frac{1}{n}\int_{0}^{1}e^{\frac{x^2}{n}}x^4\,dx \leq \frac{e}{n},
which implies that the first limit is 0 by the squeeze theorem.

I'm not sure how to compute the second limit though. I can find a lower bound on the integrand via e^x > x + 1, but what is a suitable upper bound on the integrand? Thanks in advance.

[tiny-tim]

I'm not sure how to compute the second limit though. I can find a lower bound on the integrand via e^x > x + 1, but what is a suitable upper bound on the integrand? Thanks in advance.

Hi snipez90! :smile:

Have you tried expanding ex2/n - 1 as a function of x2/n ?

[snipez90]

...

\frac{x^2}{n} \leq e^{\frac{x^2}{n}} - 1 &=& \frac{x^2}{n} + \frac{x^4}{n^2}\cdot\frac{1}{2!} + \frac{x^6}{n^3}\cdot\frac{1}{3!} + ... \\
&=& \frac{x^2}{n} + \frac{x^4}{n^2}\left(\frac{1}{2} + \frac{x^2}{n}\cdot\frac{1}{3!} + \frac{x^4}{n^2}\cdot\frac{1}{4!} + ... \right) \\
&\leq& \frac{x^2}{n} + \frac{x^4}{n^2}\left(\frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ...\right) \leq \frac{x^2}{n} + e\cdot\frac{x^4}{n^2}

for x in [0,1]. Hence

\frac{1}{3} \leq n\int_{0}^{1} e^{\frac{x^2}{n}}\,dx \leq \frac{1}{3} + \frac{e}{5n},

from which it follows that the limit is 1/3 as n -> infinity by another application of the squeeze theorem, which should be the answer to the original limit.

Thanks tiny-tim.