A great way to remind a new client of how many sessions they have left is to have them sign off each time you train with them, and show them how many sessions they have left. You may already be doing this, and it's something gyms typically do, but I have also used it in my in-home training business. Like it was said above, if your client is rockin' and rollin', and seein' progress, they will obviously WANT to renew and it won't be an issue if they can afford it.
Also, prior to or during that "last" session, re-assess them..."show" them their progress. Get them talking about how they've improved, etc. Have them set new goals for the upcoming weeks. If you have been a key factor in their success, they would be crazy not to renew!
When I worked for gyms, i'd usually hint around that "our time is almost up" and 9 times out of 10 i'd immediately know if that client would renew or not. Most of the time i'd know way before that point if they were going to renew or not. You can sense it. If you've built a relationship and "clicked" with that client, they aren't going anywhere...even if they don't train with you as often, they will still train with you...heck, even if it's a year down the road. It happens!
I have found the best way to weed out the "short term" clients is to let them know up front that it's not going to be an overnight process, etc. If it's a financial issue you'll know. I haven't had to "renew" anyone for a while now because I have a core group of people who pay me regularly...i've begun to feel like i'm a part of their family/life and that's pretty cool.
Going to peoples' houses is different than having them come to you, and I feel i'm able to build a better relationship with these types of clients.