As the school year builds towards its climax (i.e. exams), I feel almost overwhelmed with work, school, and my upcoming trip to California. There is a lot to do in the next few weeks, and it seems not enough time to do it! I won't get into the specifics of my personal life though, since this is after all the course log for CSC165.
One consequence of my monolithic work load is that the current assignment in this course seems more intimidating than ever, especially the last two questions. Calculus was always my worst subject in high school (and at U of T), so I anticipate quite a struggle in solving these. I have the first three questions done, but the nature of the last two feels alien to me, and I have avoided even trying to solve them for 24 hours now, using my work in other courses as an excuse. However, with the assistance of the course notes and my black belt in Google-fu, I am confident that I will be able to work out how to solve both problems before Wednesday night.
My primary concern is with question #4, where you have to make use of limit techniques to prove a statement involving variable powers. I barely managed to figure out how to use the definition of a limit to solve question #3, so I anticipate it will be even tougher to wrangle the answer out of question #4.
To start, I would have to figure out whether I need to prove or disprove that 3^n is in O(2^n). Intuitively, it seems that the claim is true, since both 3^n and 2^n are within the same order n, so some value c should be able to make 2^n greater than 3^n. Therefore, it would seem that (1/c)(3^n/2^n) would always be less than 1. This is about as far as I have gotten on creating an action plan to solve the problem, and I don't yet know where limits come into play. Hopefully looking at it tomorrow with fresh eyes will yield some epiphany.
