hello, i have a 2006 lexus es330. i removed the air filter housing on my car and noticed there was a line coming out of the transmission and the end of the line was broken off. the ends are plastic and im not able to remove the broken end from the hole inside the transmission. what does that line do? should i replace the part asap? can i live without it or will it cause problems down the line? added picture for reference. thanks

That is a vent. It allows pressure to escape the transmission, and is very important.
the end of the line was broken off.
which end? The top is supposed to be open and there's a little checkvalve there.
If the part that enters the transmission is broke, then replace it so you don't get water in there
That looks like a tv (throttle valve) cable. Transmissions equipped with those use the cable as a throttle position input to control internal pressure, shift points, and downshifts. If not working properly the transmission can be severely damaged. If that cable is broken it needs to be replaced immediately, and proper adjustment is critical.
TV cables are a thing of the distant past. Replaced by serial data.
Some vehicles used them into at least the early 2000s, even with otherwise electronically controlled transmissions. This video shows adjusting the cable (here referred to as a "kickdown cable") on a 2006 Toyota:
https://www.youtube.com/watch?v=IV9_-_dqa44
that's still 20 years ago, and even then I'd say they weren't common, especially in N.America. Toyota kept them around a little bit longer in international Land Cruisers. Probably deliberately to keep them more low-tech.
