i purchased my apple care services through my phone company but when i check it on the support page it only shows limited warranty. how long will it take to show on the apple support page?

I believe if you purchase from the carrier, you have to take your receipt and activate it yourself. I don't think it is done automatically like it is when you make a purchase directly through Apple. You can use the Contact Support link at the top right of the page here to check with Apple on that.

I purchase AppleCare on a MacBook Pro from an Authorized Reseller at the same time as I purchased the MacBook Pro and it took several weeks (possibly a month) before it showed on the support page. It was done automatically.