calculus - Definition of derivative

Well, I know that the derivative of a function $f(x)$is defined this way:

$$\frac{df(x)}{dx} = \lim_{\Delta x\to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$

And it's pretty clear that the expression inside the limit will approach the tangent line at a given point. I know that this is the definition of derivative. However, we can't define this to be equals the tangent line at a given point. So how do we know that this limit will in fact be equal the slope of the function?