Algebra vs. Calculus

1. In a discontinuous function, finding the limit of a certain point (c) and plugging in (c) are really different. Plugging in (c) will give you an exact output, while finding the limit of (c) will give you the y value as the graph approaches (c). If you plug in (c) into a discontinuous function, there may not be an output, if there is a hole. If you take the limit you will get an answer because i the limit is not exactly at the point of discontinuity but it is relatively close.

If the Function is continuous, it doesn't really matter if you plug in (c) or take the limit, because either way you will get the same output. The only way this can be possible is if there are no holes on the graph.

2. Slope and Derivative is basically the same thing. They both give the rate at which the graph is traveling. All they are is the change of y over the change of x.

Slope and derivative also have their differences. Slope is what is used for lines an only lines. Derivatives are for more complex graphs like parabolas and other graphs besides lines. The Derivative of a graph requires both a secant line (a line that hits the graph at two points) and a tangent line (a line that hits the graph at one point). The Derivative also requires the use of limits. Derivatives are far more complex than slope, it also requires more work to find.