Chapter 8.11, Problem 12P

Evaluate the functions $f_n(x-a)$ defined by the integral in (11.13) with limits $-n,n$.

Show that ${\displaystyle {\int }_{-\infty }^{\infty }{f}_n}(x-a)dx=1$ for all n. Sketch or computer plot graphs of several $f_n$ ’s to show that as $n$ increases the functions $f_n(x)$ are increasingly peaked around $x=a,$ and that as $|x-a|$ increases, they oscillate with decreasing amplitude.