# Is this step mathematically allowed?

In my signal and systems course, my professor claims that the energy of a signal $$x(t)$$ over an infinite and continuous time interval is: $$E_{\infty}=\lim _{T \rightarrow \infty} \int_{-T}^{T}|x(t)|^{2} d t=\int_{-\infty}^{+\infty}|x(t)|^{2} d t$$ My concern is about the substitution of $$\infty$$ in place of $$T$$ before evaluating the integral. Is this always correct or is it mathematically more correct to evaluate the integral then compute the limit?

• Look back at the definition of $\int_{-\infty}^{+\infty}$ - there's less here than meets the eye! Commented Feb 2, 2021 at 4:54
• Commented Feb 2, 2021 at 4:55
• Thank you so much I think my question was embarrasing and hence I will delete it oh but before I do, I noticed in the wikipedia page that they said this is an abuse of notation to replace the upper bound with a $\infty$ Commented Feb 2, 2021 at 4:57

Many people write $$\int_{-\infty}^{\infty} |x(t)|^2dt$$ as $$\lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt$$ because they believe that using infinity as a number is not proper, so they write it as a limit. However, since $$\int_{-\infty}^{\infty} |x(t)|^2dt = \lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt$$
is true, that must mean that the converse, $$\lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt =\int_{-\infty}^{\infty} |x(t)|^2dt$$