Hi GT,

To a certain extent, the slopes are assumptions. But they are also in a way the real values. Much of calculus is based on theory and concepts. The derivative (a limit) is a theoretical concept used to find the exact slope of a function at a specific point. Normally, you would need two points to find a slope (using a secant line). But when you use the derivative, you can find the slope using only one point. 

When looking at the h-form version of the derivative as a limit,

the numerator is finding the change in y-values, and the denominator is the change in x values. This is the same concept that you would use to find a slope using two points (using a secant line). And to a certain extent, you can visualize this limit as taking two points on a curve, and sliding them closer and closer together until the distance between them on the curve gets closer and closer to 0. Eventually the points get so close together that the distance between them is irrelevant, and you could almost think of them as one point, but you can still use those two points to calculate the slope. By taking the limit of the approximate slopes (slopes of the secant lines), you are able to find the exact slope (using the tangent line).

By using limits (specifically the derivative) we are able to make the "assumed slope" arbitrarily close to the real slope at the point. We are able to turn an infinite number of calculations into a finite conclusion.

Let me know if you have any questions! Hopefully I didn't confuse you : )

Hey Evan, I think my question could have been reworded to simply ask if there was a method were instead of calculating a limit as it approaches a value, but at the value itself. Reading your explanation I understand that while we make a lot of assumptions in the mathematics itself, it is because we must.  It is not possible to calculate values to infinity, and so we have created a method which allows us to get as close of an approximation to it as we can.  Anyways thanks for taking the time to reply, and explain.

GT

Hi GT,

I think I have an answer to your question, if I'm understanding what you're asking correctly.

A "regular" limit (not a derivative, integral, etc.) finds the value that f(x) approaches as x approaches some value. Based on the definition of a limit, we are calculating the value f(x) is approaching as x approaches the value, but x never becomes equal to the value.

To answer your question, yes there is a way to find the value of f(x) at the exact x-value, but it doesn't involve limits. To find f(x) at the exact x-value, you evaluate f(x) at x (plug the x-value into f). However, this only works when f(x) is defined at that x-value (the curve has no holes). 

This method only allows you to find the value of f(x) at that x-value, not the slope. To find the slope, we have to use limits. Once you get to differentiation rules however, you will see that once you find the derivative of a function, you can simply plug the x-value into the derivative to find the slope. In this sense, you are finding the slope at that exact x-value, rather than for values really close to the x-value.