I'm from South Korea. We change ㄴ/ㄹ to ㅇ when it comes first. But when it comes to one's name, we don't care because it's his/her name. One of famous Korean athletes is a baseball pitcher 류현진. He don't like to be called 유현진. So you can use 임 or 림 as you like. But actually it's natural for us to use 임, so don't be embarrassed when people hear your name and are puzzled. Hahaha In North Korea, there's no change. So they use 림 only. hahaha

I'm Korean native. I can't believe what the above people are telling you. If your last name is spelled Lim, it should be written as 임, not 림 in Korean. Yeah, sure, people in North Korea might write as 림, but South Koreans never do that.

As your last name Lim is a foreign language, its Korean spelling is 림.

Lim is written 림