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Homework Statement
Show that [tex] \delta_n(x) = ne^{-nx} \quad \mathrm{for}\quad x>0[/tex][tex] \qquad = 0 \quad \mathrm{for}\quad x<0[/tex]
satisfies [tex]\lim_{n\longrightarrow\infty}\int_{-\infty}^\infty \delta_n(x)f(x)\mathrm{d}x = f(0)[/tex]
The attempt at a solution
The hint says to replace the upper limit ([itex]\infty[/itex]) with [itex]c/n[/itex], where [itex]c[/itex] is "large but finite", and then use the mean value theorem of integral calculus.
I do not understand how this replacement in the hint is allowable. Since [itex]n\longrightarrow\infty[/itex], [itex]c/n\longrightarrow0[/itex], not [itex]\infty[/itex]. Even if this is okay, how does it aid using the mean value theorem of integral calculus?
Show that [tex] \delta_n(x) = ne^{-nx} \quad \mathrm{for}\quad x>0[/tex][tex] \qquad = 0 \quad \mathrm{for}\quad x<0[/tex]
satisfies [tex]\lim_{n\longrightarrow\infty}\int_{-\infty}^\infty \delta_n(x)f(x)\mathrm{d}x = f(0)[/tex]
The attempt at a solution
The hint says to replace the upper limit ([itex]\infty[/itex]) with [itex]c/n[/itex], where [itex]c[/itex] is "large but finite", and then use the mean value theorem of integral calculus.
I do not understand how this replacement in the hint is allowable. Since [itex]n\longrightarrow\infty[/itex], [itex]c/n\longrightarrow0[/itex], not [itex]\infty[/itex]. Even if this is okay, how does it aid using the mean value theorem of integral calculus?