Algebra vs. Calculus

1. When you're finding the limit of a function at x=c, you're finding the value that f(x) approaches as x approaches some constant. When plugging in x=c into f(x), you're finding what the actual value of f(c) is, even if f(x) doesn't necessarily approach f(c). The two cases are the same when f(x) is continuous at x=c, as given by the definition of continuity at a point lim f(x) x->c=f(c).

2. The similarities between finding the derivative and the slope of a line is that you essentially use the same formula (m=change in y/change in x), the only slight difference being when finding the derivative, you bring the two points infinitely closer to each other so that they become the same point. Mathematically, for the derivative, you use the formula lim (change in y/change in x) h->0, h being the distance between the two points. The difference is that while they are both the slope of a line, one is the slope of a specific line while the other is the slope of any line. When finding the derivative, you are finding the slope of a line, but you are finding the slope of a tangent line to a point on a curve. When saying you are finding the slope of a line, it generally refers to any ordinary line. In other words, when saying you're finding the slope of a line, it does not tell us any of the line's special properties, if it has any.

Limits

Eh, the last post was better. Anyway, for the most part, I understand limits very well. There's just one or two things that elude me:

1. The first would be the limit of the basic trig functions (sin x, cos x, tan x, csc x, sec x, cot x) as x-> positive or negative infinity. The problem is that unlike most functions, who reach a specific number or positive or negative infinity, the trig functions don't. They simply alternate between [-1,1] (sin x and cos x), (negative infinity,-1] and [1, infinity), (csc x and sec x), or (negative infinity, infinity) (tan x and cot x). Does that mean that the limit simply does not exist?

2. #13 and #14 on page 92 also confuse me. I know what they're asking and how to generally find what they're asking for, but when I try to find it algebraically, I get something that can't be simplified. Because they're absolute value problems, you can figure out the slope from the graph, but how would you find it algebraically?They are:

Find the slope of the curve at the indicated point:
13.f(x)=absolute value (x) at: a)x=2 b)x=-3
14.f(x)=absolute value (x-2) at x=1

Hm....yea, that's about it, I don't have a third thing: that's it. If anyone has answers to these questions, I would greatly appreciate it.