How do limits apply to $y=x^2$?

Question

The full derivative of the function $y=x^2$ is: $$y'=2x+\varepsilon$$ Calculated as follows: $$y'=\frac{(x+\varepsilon)^2-x^2}{\varepsilon}=\frac{x^2+2\varepsilon x+\varepsilon^2-x^2}{\varepsilon}=2x+\varepsilon$$ If we were to 'take the standard part' we would just drop the $\varepsilon$ so as to get the result for the tangent (rather than a secant) but if we were to reason with limits we would have to say something like "since we can reduce $\varepsilon$ as low as desired $y'$ can get as near as we like to $2x$". How can that be put using the normal symbolism? That is, instead of defining limits using general variables and functions can we define them for this case first? And then generalize after.

Answer 1

Great question! And also one that reveals a common misconception, namely the idea that limits are somehow a different approach than the one with infinitesimals (and standard part). In fact, it is the same approach, in the sense that limit is expressed in terms of the standard part. Thus, if $y=f(x)$ then the derivative of $f$ at $c$ can be defined as $\lim_{x\to c}\frac{\Delta y}{\Delta x}$ where $\Delta x$ is an ordinary real independent variable, but it can also be defined as $\textbf{st}\left(\frac{\Delta y}{\Delta x}\right)$ where $\Delta x$ is an infinitesimal independent variable. More generally, $\lim_{x\to 0} f(x)= \textbf{st}(f(x))$ where $x\not=0$ is infinitesimal, whenever the limit exists.