If I take Pre calculus in high school will I be eligible to take classes that require Math 105 (Pre Calculus) in college?
Yes, in theory. However, many colleges have an advisory or mandatory math placement exam to check that (for example) students who want to take calculus know their precalculus material. If your high school precalculus course is of high quality and you do well in it, you are likely to do well enough on such a math placement exam to take calculus.
Examples of math placement tests:
http://www.math.buffalo.edu/rur_index.html
http://math.tntech.edu/e-math/placement/
https://math.berkeley.edu/courses/choosing/placement-exam