## Limits and the Infinitesimal Number

I’ve been thinking about the infinitesimal number, $\delta = \frac{1}{\infty}$, recently. In particular, that one could use it to evaluate limits.

What is a limit, really? I’ve been reading some topology recently and I think that it really is a function that returns an accumulation point (hint: these are alternatively known as limit points). More specifically, I believe that $\lim_{x\to a} f(x)$ is an attempt to find a value $y$ such that $(a,y)$ is a limit point of the graph of the the function $f$.

But I’ve digressed since the simpler, “It’s the value as we approach the point” is perhaps more useful to us…

Consider $\lim_{x\to a} f(x)$. How is this defferent from $f(x\pm\delta)$? The difference is that we’re looking for the hypothetical value that the function is becoming (also the value of the point which any open set containing it intersected with the graph is not null), not its value when it is infinitly close. Consider $\lim_{x\to 2} x$: the difference is $2$ versus $2\pm \delta$. So, we need to get rid of the infintesimal difference. Let $\mathbb{R}(x)$ represent the rounding of anumber $x$ to the nearest real number. Then,

$\lim_{x\to a} f(x) = \mathbb{R}\cdot f(x\pm\delta)$

Does this have any applications? I beleive it may provide a more elegant way to present Calculus.

Why can’t I take the first principles defenition of a derivative and:

$\frac{dy}{\rlap{---}dx} = \lim _{h\to 0} \frac{f(x+h) -f(x)}{\rlap{--}h}$

? Because the defenition of a limit would give us zero, that being the closest real number. But instead, we could just say that a value is close to delta and use that. Not useing $h$ because I see no reason to use a random symbol when there is a logical one.

$dy = f(x+dx) -f(x)|_{dx\simeq\delta}$

And we have a differential form, let’s add vectors (for more dimensions):

$dy = f(\vec{x}+\vec{dx}) -f(\vec{x})|_{|\vec{dx}|\simeq\delta}$

and for yet more clarity:

$dy = y(\vec{x}+\vec{dx}) -y(\vec{x})|_{|\vec{dx}|\simeq\delta}$

which is far less cumborsome than

$l\lim_{\vec{dx \to 0}} \frac{f(\vec{x}+\vec{dx}) -f(\vec{x})}{|\vec{dx}|}=0$

And has almost identical properties (I suppose that since it is $+ \delta$ it will be forward facing, eg, $d|x|(0) = \delta$ instead of undefined).

Just some random thoughts.

$dy = f(\vec{x}+\vec{dx}) -f(\vec{x})|_{|\vec{dx}|\simeq\delta}$
The most grievous error is that the normal differential definition is $\lim_{h\to\vec{0}}\frac{f(\vec{x}+\vec{h})-f(\vec{x})-dy}{|\vec{h}|}=0$.