A common use case for Web Designers and Developers to need to figure out a client's old IP address after maybe not planning for it, would be something like trying to bring back a clients old website to take a look at the content after pointing the A Record to the new site. 

Or, if you're like me, a couple weeks ago, I actually went in during the middle of the night to migrate a client's website, and pointed the WRONG A RECORD! Whoops. 

I needed to retrace my steps, but hadn't documented what I was doing. So I needed to figure out how to get the old IP address. 

Thank goodness I found SecurityTrails.com and brought their website back up immediately, as shown in this video.