Let \(g(x)=\exp (-|x|)\) and define a sequence of functions \(\left\{f_{n}(x)ight\}_{n=1}^{\infty}\) as \(f_{n}(x)=g(x) \delta\{|x| ;(n, \infty)\}\), for all \(n \in \mathbb{N}\).

a. Calculate

\[f(x)=\lim _{n ightarrow \infty} f_{n}(x),\]

for each fixed \(x \in \mathbb{R}\).

b. Calculate

\[\lim _{n ightarrow \infty} \int_{-\infty}^{\infty} f_{n}(x) d x\]

and

\[\int_{-\infty}^{\infty} f(x) d x\]

Is the exchange of the limit and the integral justified in this case? Why or why not?