Why $\int_0^{\infty} f(t)dt = \lim_{\tau \to \infty }g_\tau (0)$?

Question

It is stated (click here and go to theorem 16.13 on page 7),

By definition, $\int_0^{\infty} f(t)dt = \lim_{\tau \to \infty }g_\tau (0)$

The image is given below-

Please show elaborately why it is true, thanks.

Answer 1

Just go by definition.

$$ g_\tau(0) =\int_0^\tau f(t)e^{-0\cdot t} dt$$

The exponential term $e^{-0\cdot t}$ is equal to $1$. So,

$$ g_\tau(0) =\int_0^\tau f(t) dt$$

Now apply the limit

$$ \lim_{\tau\to\infty}g_\tau(0) = \lim_{\tau\to\infty}\int_0^\tau f(t)dt$$

$$ = \int_0^\infty f(t)dt $$