To fix a slow Limewire connection you need to uninstall the Limewire software and redownload the new version of Limewire.
Because of your internet connection
You can't "fix the connection to Minecraft" it's all on your current holder, and bandwidth.
It is a troubleshooting process beginning with the easiest and culminating in the hardest (i.e., most expensive). It can be an interruption in service or a profile corruption, or slow connection.
It lags with slow connection , bad connection or corrupted connection
You might want to try clearing your cache or checking your internet connection. Otherwise, your computer may have caught a virus or is normally slow due to older versions or incompatible versions of Adobe Flash. Try upgrading flash if clearing your cache does not work and/or your internet connection is fine.
Get a faster Internet connection.
Sometimes, the only thing a person can do to fix a slow Windows 2000 computer is to reset it to factory defaults. Perhaps the computer is running slow because the memory is used up. In this case, delete things that are not necessary on the computer or download them to a disk.
No one likes a Windows XP slow startup. When you startup your computer you want it to be ready to use immediately. I can not make it that fast, but I can help you fix slow windows xp. Read more: http://slow-xp.com/how-to-fix-slow-window-xp-and-optimize-your-computer.html
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After I update windows my browsing connection is slow even if my internet speed is fast with other PC connected.
You can test your Xbox Live Connection on your dashboard under "My Xbox" and if there is an issue with the connection it will tell you what it is.
Your internet can be not faster than your internet connection (DSL, cable modem and so on). If you have a slow internet connection, installing a new router will not fix the speed. It might improve stability of wireless signal and security though.