Recently I bought a fairly used Samsung s3 AT &T model. And I've been having problems keeping it on constant 3g or changing it to edge when my battery is down. Since it's not in the phone's settings. But it all ended today as I found a way to solve the problem.
Type in this code in your dialer
*#*#4636#*#*
Then select Device Information
Scroll down, and you'll see
GSM/LTE/WCDMA
Click on it.
Then select anyone you which.
For me I prefer WCDMA only when I need high speed internet since there's no 4g in my area. But if there's 4g is your neighborhood, select WCDMA(3.75G)/LTE(4G) just in case it steps out of LTE(4G).
But if you want to save your battery, select GSM or EVDO.
Before you even attempt any of the above, make sure you data is turned off.
After you must have changed it, exit, then turn on data.
It's that easy.
Thank me later.

Screenshots

Nice share.

guyX:

I have same problem before now, but have resolved it.
The issue now it that my flash share and xender can't create connection. Any time I try to do that it always request I switch on mobile data.



And secondly my mobile Hotspot won't switch on too.

Joejonah:




I have same problem before now, but have resolved it.
The issue now it that my flash share and xender can't create connection. Any time I try to do that it always request I switch on mobile data.




And secondly my mobile Hotspot won't switch on too.

You need to have your phone rooted for starters.
Your hotspot and Wi-Fi is linked to AT & T sim card. That's why it says insert sim.

Mine does that too but when I turn on xender, it(hotspot and WiFi) turns on.

Google XDA-developers forum and the issue.