Why can the length of vector $r(t+dt) - r(t)$ be approximated with the vector $r'(t)dt$? **Question:** This passus was presented in my textbook. I can't understand their reasoning. Why can the length of vector $r(t+dt) - r(t)$ be approximated with the vector $r'(t)dt$? **From my textbook:** Let $r(t) = \left( x(t), y(t)\right)$. We can then approximate the curve length between the points $r(t)$ and $r(t+dt)$ with the length of the vector $r(t+dt) - r(t)$. We can in turn approximate this vector with the length of the vector $r'(t)dt$, i.e: $$|r'(t)|dt = \sqrt{x'(t)^2 + y'(t)^2}dt$$ ![enter image description here](

Because, by definition, $$r'(t) = \lim_{dt\to 0}\frac{r(t+dt)-r(t)}{dt}$$ This means that, for very small $dt$, $r'(t)\approx\frac{r(t+dt)-r(t)}{dt}$; that's just how one interprets limits. In other words, $r'(t) dt \approx r(t+dt)-r(t)$.

As a similar use of this reasoning: $e=\sum_{n=0}^\infty\frac{1}{n!}$, and so $e\approx \sum_{n=0}^N \frac{1}{n!}$ for very large $N$.