Why does $h$ go to $0$ in the first derivative?

Question

1st yr economics student and currently doing calculus for the first time.

In the first derivative, does the limit of $h$ or delta $x$ always go to $0$? Is this a general rule or does it depend on the curve? For example I'm assuming all upward sloping curves have $h\to0$, would a downward sloping curve therefore have $h\to$ infinity when going towards the $x$ axis?

Also why does this happen? I'm guessing its because to find the gradient between the higher point of a curve and the lower point, you would travel towards the origin and therefore $h\to0$; is this correct or have I misunderstood?

Very confused at the moment as to whether the first derivative is universal or depends on the properties of the curve.

Answer 1

You have misunderstood. Intuitively the idea is that we are trying how much the function changes in a small interval around a point $x_0$. What we do is pick points close to $x_0$ and calculate the slope of the tangent line. Those are points of the form $x_0 + h$ where $h$ is a small quantity. To find the behavior in $x_0$ we let $h \to 0$, this is, we explore points infinitely close to $x_0$. This is independent of the function, it’s just the way we approach the point we want to explore.

Answer 2

Source

In the above diagram, the blue curve is the function $f$, and the red line approaches the tangent to the curve. The derivative $f'(c)$ is $$\lim_{h\to0}\frac{f(c+h)-f(c)}{h}.$$

Regardless of the shape of the (differentiable) curve, as $h$ (the gap between the orange and red points) goes to zero, the two points get closer to each other, and the red line gets closer to being a tangent to the blue curve, which means that the gradient of the red line gets closer to the derivative of the blue curve at the point where $x=c$.

We are in essence taking $\dfrac{\text{rise}}{\text{run}}$ as $\text{run}$ approaches $0$. This will give us the gradient as a point, instead of between 2 points.

It has nothing to do with upward sloping or downward sloping, nor are we necessarily finding "the gradient between the higher point of a curve and the lower point", we just want the gap between the two points on the curve to go to zero so that we end up with a line that is tangent to the curve.