How does using limits to find the derivative of a function avoid dividing by $0$?

Question

What is wrong with this argument that differentiation using the First Principle leads to division by $0$:

$$ f'(x)=\lim_\limits{h \to 0} \frac{f(x+h)-f(x)}{h} \\ $$

Using the quotient limit law:

$$ \lim_\limits{h \to 0} \frac{f(x+h)-f(x)}{h}=\frac{\lim_\limits{h \to 0}f(x+h)-f(x)}{\lim_\limits{h \to 0}h} $$

$$ \lim_\limits{h \to 0}h = 0 $$

Therefore, there the top half of the fraction is divided by $0$. Here is my reasoning for why $\lim_\limits{h \to 0}h = 0$:

As $h$ approaches $0$, its value becomes smaller (and will become smaller than any number strictly greater than $0$). For example, you cannot evaluate the limit as equalling $0.001$, because at some point $h$ will be lower than this. $0$ is the largest number that does not have this problem. Therefore, the limit expression is equal to $0$.

Thank you for reading.

Answer 1

The part which is wrong in your reasoning is this: $$\lim_\limits{h \to 0} \frac{f(x+h)-f(x)}{h}=\frac{\lim_\limits{h \to 0}f(x+h)-f(x)}{\lim_\limits{h \to 0}h}$$

The formula $\lim \frac{f}{g}=\frac{\lim f}{\lim g}$ can only be used when $\lim(g)\neq 0$. If $\lim g \neq 0$ the RHS of your formula, does not make any sense.

To understand what happens here, just look at $$\lim_{h \to 0} \frac{h}{h}$$

$\frac{h}{h}=1$ and $\lim_1=0$. But you cannot write $$\lim_{h \to 0} \frac{h}{h}=\frac{\lim_{h \to 0} h}{\lim_{h \to 0} h}$$ since the RHS makes no sense.

Answer 2

You can only use the quotient limit law when the limit in the denominator is not $0$. As you have said, both limits are $0$, so you cannot use it. $h$ is explicitly not $0$ in the limit definition. What you usually do is find a way to divide out the $h$ in numerator and denominator analytically. Then the limit of numerator and denominator are nonzero, in fact the denominator is a fixed $1$, and you can use your limit law.

Answer 3

1) The quotient limit law can be used when the limit exist and the one at the denominator is nonzero. It is not the case in your example.

2) When you compute a limit, you never evaluate the function at the limit point, only in neighborhoods. So in your example, $h\ne0$.

A limit is "the natural analytic continuation" of a function at a given point.