As aforementioned, I have thoroughly struggled through my college-level Calculus II class, even though I aced my High School AP Calculus course, and even got a 5 on the AP exam. My professor assigned the following seven extra-credit problems, all of which baffle me. I am unsure of how much extra credit each problem is worth, however I am wanting to put forth as much effort as possible in order to pass the course, even if with a C. I uploaded the questions to tinypic, at the following link:

http://i44.tinypic.com/2n7qv4l.jpg

I would appreciate any input on how to solve the following problems. I am not looking for answers (even though they would be GREATLY appreciated), but even any suggestions to get me on the right track would be so helpful, as it might make the difference between my passing or failing of the course.

[Moderator's Comment: Hints and pointers only will be allowed for these, infractions will be given for any detailed or full solutions]

Edit: An absolutely wonderful individual gave me these pointers for the first three problems. Let me know what you guys think.

1) I am looking for functions that when added together never settle down into a limit. I.E., an oscillatory function. In order for the individual functions to grow without bound we multiply by a factor of x or -x to push it the right way. In order for their sum to result in just the oscillatory function we need to offset the 2nd function.

We add two to the sine because we don't want the factor to ever be zero. This would make the limit not exist for f and g.

f(x) = (x)*(sin(x)+2) , as x→∞ , f(x)→∞

g(x) = (-x+1)*(sin(x)+2) , as x→∞ , f(x)→ -∞

But f(x) + g(x) = sin(x)+2 , which has no limit as x→∞

2) Some ideas:

Does n represent an integer? If considered as a real number the limit wouldn't exist (I think). For most values of sin the function tends to 0, but no matter how large n we can always find a value where the function is 1 or -1. Hence non-convergence.

***Note: I believe that n represents any real number, not just an integer.*** - third.beyond

Infinite limit is defined to be :

for all ε>0 there exists an S such that |f(x)-L|<ε whenever x>S

So for ε<1 no matter the S there is an x such that the inequality doesn't hold (when the sine is ±1).

If the limit is only over integer values: then I'm not quite sure about the limit. No matter what integer value we put in, the sine function won't be 1 or -1 since π/2 + 2cπ or -π/2 + 2cπ for any integer c is not an integer (π is transcendental). This makes me think the limit exists and is zero.

3) This question leads to something very interesting. This type of recurrence relation gives rise to a unimodal mapping typical in introductory chaos theory courses. Look at the logistic map on wikipedia for another example. You can prove some convergence stuff using linearization of the mapping. The fixed point converges through damped oscillations. You can show this graphically by plotting successive mappings (called a cobweb map)

http://en.wikipedia.org/wiki/Cobweb_plot

Otherwise you can use the results from linearization to talk about stability, and convergence behavior.

**Let me know what you guys' think about those proposals