Algebra vs. Calculus

1.The difference between finding the limit of a function as x=c and actually plugging in the number x=c, is that the limit of a function as x approaches c could not be a value of the function if the function is not continuous at c. In a limit you are finding the value as x is approaching the constant c which can be different to what the actual value is. When plugging in x=c to the function you are finding the exact value of the function. The best way to see the difference is through a function that is not continuous for example a piecewise function which can have several different functions with different limits when x=c or a different value when x=c.
The two cases are the same when the function is continuous were the limit when x approaches c is the same as the value of the function when x=c.

2. The similarities when finding the derivative of a line and the slope is that the value is the same in the case of a line because in both functions you are finding the change in y over the change in x. In a derivative when finding the value we use limit as h approaches 0 f(a+h)-f(a)/h meaning that we are assuming that the difference as h approaches a is infinitely small.
Derivatives are mainly used to find the slope of a point in a curve by finding the slope of its tangent line.