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.