Clarification of Notion of a "Good Approximation"

Question

My textbook says the following:

$$\lim_{x \to x_0} \dfrac{f(x) - f(x_0) - f'(x_0)(x - x_0)}{x - x_0} = 0$$

Thus, the tangent line $l$ through $(x_0, f(x_0))$ with slope $f'(x_0)$ is close to $f$ in the sense that the difference between $f(x)$ and $l(x) = f(x_0) + f'(x_0)(x - x_0)$, the equation of the tangent line, goes to zero even when divided by $x - x_0$ as $x$ goes to $x_0$. This is the notion of a "good approximation that we will adapt to functions of several variables, with the tangent line replaced by the tangent plane.

I'm finding it difficult to understand how this notion of "good approximation" makes logical sense, given logical reasoning and all of the other mathematics I've learned. For $\dfrac{f(x) - f(x_0) - f'(x_0)(x - x_0)}{x - x_0}$, as $x \to x_0$, we have that the numerator and denominator are approaching $0$ at the same rate -- after all, there are no exponents to indicate that one is approaching $0$ quicker than the other. Given this, we would usually say that $\lim_{x \to x_0} \dfrac{f(x) - f(x_0) - f'(x_0)(x - x_0)}{x - x_0} = \dfrac{0}{0}$, which is an indeterminate form. In order to have that $\lim_{x \to x_0} \dfrac{f(x) - f(x_0) - f'(x_0)(x - x_0)}{x - x_0} = 0$, we would require that the numerator approach $0$ quicker than the denominator, which, as I said, there is no indication of.

So can someone please clarify this notion of "good approximation" and explain why it makes mathematical sense? I'm also wondering if this is just a crude/"hand-wavey" way of explaining the notion of "good approximation", since, as I said, it doesn't seem very sensible, and a more sensible and rigorous way would use epsilon-delta notions? If anyone has a better mathematical explanation of the notion of "good approximation" feel free to share.

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

You are right that in general the numerator need not approach zero faster than the denominator.

But the point here is that IF the numerator does approach zero faster than the denominator , THEN we say that $f(x_0) + f'(x_0)\, (x-x_0)$ is a good approximation to $f(x)$ as $x \to x_0$.

There is no hand-waving at all involved here. This is the definition of what the phrase “good approximation” means in this context, and it's all based on the $\epsilon$ & $\delta$ stuff built into the rigorous definition of the concept of limit.

Answer 2

You can write a Taylor series for $f$ expanded around $x_0$, which gives $$f(x) \approx f(x_0)+(x-x_0)f'(x_0)+\frac 12(x-x_0)^2f''(x_0)+\ldots$$ When $x$ is close to $x_0$ the factor $(x-x_0)$ is small, so as long as the derivatives do not get too large the terms in the expansion will decrease. You can cut off the sum when the terms get small enough. As an example, let $f(x)=e^x, x_0=1$, were we know all the derivatives are $1$ as well. If we want $f(1.01)$ we would write $$f(1.01) \approx e+0.01e+\frac 12\cdot 0.01^2e+\ldots$$ Each successive term in the series decreases by a factor larger than $100$ so it is reasonable to ignore all the terms after the first two. We say that the tangent line is a first order approximation because close to $x_0$ the error of the approximation is proportional to $(x-x_0)^2$, or quadratic.