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Show that we cannot interchange a limit and integral for fn(x) = nxe-nx2

by RoRi

For each positive integer n define

    \[ f_n (x) = nxe^{-nx^2} \qquad x \in \mathbb{R}. \]

Prove that the following limit and integral cannot be interchange:

    \[ \lim_{n \to \infty} \int_0^1 f_n (x) \, dx \neq \int_0^1 \left( \lim_{n \to \infty} f_n (x) \, dx \right). \]

Proof. First, we have

    \begin{align*} \lim_{n \to \infty} \left( \int_0^1 f_n (x) \, dx \right) &= \lim_{n \to \infty} \left( \int_0^1 nxe^{-nx^2} \, dx \right) \\[9pt] &= \lim_{n \to \infty} \left( -\frac{1}{2} e^{-nx^2} \Bigr \rvert_0^1 \right) \\[9pt] &= \lim_{n \to \infty} \left( -\frac{1}{2} \left( e^{-n} - 1 \right) \right) \\[9pt] &= \frac{1}{2}. \end{align*}

On the other hand,

    \begin{align*} \int_0^1 \left( \lim_{n \to \infty} f_n (x) \right) \, dx &= \int_0^1 \left( \lim_{n \to \infty} nxe^{-nx^2} \right) \, dx \\[9pt] &= \int_0^1 0 \, dx \\[9pt] &= 0. \end{align*}

Hence,

    \[ \lim_{n \to \infty} \int_0^1 f_n (x) \, dx \neq \int_0^1 \lim_{n \to \infty} f_n (x) \, dx. \qquad \blacksquare \]

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