I attached my problem as a JPEG file. The black ink is my own work, and the green ink is my teachers note's. If you are confused by the part that states these two steps prove continuity, I considered the evaluation of the function at a value and the evaluation of the limit at the same value to be one step, that's why there is two.
Well, I asked my teacher to write the proper way to write the solution to this sort of problem, but evidently she did not. So, I was wondering if someone could possibly interpret her notes, and also if someone could show me the mathematically proper way to solve this problem.