Algebra Vs. Calculus

1. Depending on the situation, there may or may not be a difference between the limit of a function at x=c and plugging in the number x=c. When finding the limit at x=c, one is looking for the number the function outputs as the input approaches c. There may be an hole at x=c, but if the function approaches that point from both sides, it is indeed the limit. When searching for f(c), one is looking for the output at the x value c. The output does not necessarily have to be the same as the limit of x=c. However, if the limit at x=c equals f(c), then the function is continuous.

Only when the function is condtinuous does the limit at x=c equal f(c).


2. When finding the slope of a line, one is finding the (change in y)/(change in x) of the entire line, not just between two pints. To find the slope of a line, one uses the formula (change in y)/(change in x). (y2-y1)/(x2-x1), another variation of this formula, is sometimes used. Finding the slope of a line typically does not require much more than counting rise over run or plugging in numbers into a simple formula.

To find the derivative of a line, one has the choice of using various different formulas, three of which involve limits, and one which does not. The typical, most basic formula is limit as h approaches 0 f(h+a)/h. When finding the derivative, one is basically finding the slope of a line at one specific point (this is why h is approaching 0).

Finding the slope and finding the derivative of given lines are similar in the way that both of them deal with finding the change in x and y.