Algebra Vs. Calculus

1. What is the difference between finding the limit of a function at x=c and actually plugging in the number x=c? When are the two cases the Same?

The difference between the anwers of the two ( finding the limit of the finction at x=c and finding f(c) ) depend on the function ( Whether it is continuous or not). When you are finding the limit of a function at x=c you are finnding the output as you are approaching that constant.

Example:
lim 1/x

x-> 0 the answer is infinity

When you are finding f(c) (pluggin in the x=c) you are looking for the exact output when x=c.

Example: (same as previews)

f(x)= 1/x when x=0

f(0)= undefined

As you can see the the two answers are different due to the fact that the function is discontinuous at that constant

They are usually the same when the function is continuous or at least continuous at that constant (x=c)

Example:
lim 2x+5....................................................................2x+5 when x=4

x-> 4 ............................................................................answer 13

answer 13

The function is continuous therefore the answers are the same

2. What are the similarities between finding the derivative and finding the slope of a line? What is the difference between the two?

the similarity between finding the derivative and finding the slope of a line is that you are taking the change of y ( the output) over the change of x (the input).

The difference between the two would be that you goal when finding the derivative is to finding the exact speed at that point, in other words you are finding the tangent line. When you are finding the slope you are just finding the average speed or change between two points, in other word the secant line.