The following problem seems very simple, and indeed I'm sure the answer isn't hard at all. I know how to prove the limits of polynomial functions, using the epsilon-delta definition of a limit, by factoring certain parts, using certain theorems about sums and products of limits, and by finding upper bounds for certain expressions. From my understanding, the basic "approach" to finding the following:
is to find a factor that puts the function in the following form:
for some numbers and , and some function . My problem is that I don't know how to go about doing so for the following problem:
Prove the following using the epsilon-delta definition of a limit:
Now, obviously this limit is "intuitively" true but I can't seem to figure out the exact definition of delta in the proof. I've done some scratch work, I know that generally this forum encourages that posters display any work they've done themselves so far, so I'll put that below. Granted, it may contain errors:
I started with this:
and made the stipulation (with no particular motivation other then finding an upper bound on something):
the next line is something that I'm not certain is "always" true. In other words, I'm not certain if it follows from the above line:
now, I know I need to somehow get to the point of having this statement:
but I'm not sure how to proceed correctly. Do I subtract from all parts of the inequality (the one three lines above) above? Or should I start by working firstly with ?
Any guidence would be much appreciated.