B. For $n \ge 1$, define functions $f_n$ on $[0, \infty)$ by
$f_n(x) = \begin{cases} e^{-x} & \text{for } 0 \le x \le n, \\ e^{-2n}(e^n + n - x) & \text{for } n \le x \le n + e^n, \\ 0 & \text{for } x \ge n + e^n. \end{cases}$
(a) Find the pointwise limit $f$ of $(f_n)$. Show that the convergence is uniform on $[0, \infty)$.
(b) Compute $\int_0^{\infty} f(x)dx$ and $\lim_{n \to \infty} \int_0^{\infty} f_n(x)dx$.
(c) Why does this not contradict Theorem 8.3.1?
