What is the relation between dx in elementary calculus and dx in differential geometry? I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is $dx$?!
As far as I understand, in differential geometry $dx^{i}$ is a linear functional that maps vectors in a tangent space $T_{p}M$ at a point $p\in M$ on a manifold $M$ to the set of real numbers $\mathbb{R}$, i.e. $$dx^{i} :T_{p}M\rightarrow\mathbb{R}$$ In this sense the differential form $dx^{i}$ maps a vector $v\in T_{p}M$ to its $i^{th}$ coordinate with respect to the coordinate basis $\frac{\partial}{\partial x^{i}}$, i.e. $dx^{i}(v)=v^{i}$.
In elementary calculus I was always told when I asked the question "what is $dx$?", that it is an infinitesimal change in the x-coordinate. This has never rested easy with me as e.g. if we have the formula $$ df=\lim_{\Delta x\rightarrow 0}\Delta f = \lim_{\Delta x\rightarrow 0}f'(x)\Delta x $$ then due to the properties of limits this can be expressed as $$\lim_{\Delta x\rightarrow 0}f'(x)\lim_{\Delta x\rightarrow 0}\Delta x$$ and clearly $\lim_{\Delta x\rightarrow 0}\Delta x =0$ which seems inconsistent.
So my main question is: what actually is $dx$ and is there any intuitive (perhaps geometric) explanation as to how it relates to an infinitesimal line element?
 A: In dimension one:
$$df = f'(x)dx$$
or better:
$$dy = f'(x)dx$$
Here $${T_x}R$$ and it's dual $$({T_x}{R) }^ *$$
are 1-dimensional vector-spaces which are identified with $R$ itself as reals.
In this case you are allowed to do divisions. 
There is no mystic behind:
$$\frac{{dy}}{{dx}} = f'(x)$$
$dx$ can be viewed as 1-dimensional volume-element, and function $f$ as change
of variables. Volume-element $dy$ with respect to $x$ has shape like $dy = f'(x)dx$
$$f'(x)$$
is the 1-dimensional Jacobian!, if you want. By the way if the limit
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}$$
at a point exists, then we have
$$\frac{{dy}}{{dx}}({x_0}) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = f'({x_0})$$ 
In general $\frac{{\Delta y}}{{\Delta x}}$ and $\frac{{dy}}{{dx}}$ are fractions, but different ones. Namely slope for secant-line and second slope for tangent-line.
And because they are reals, you can draw them in 1-dimensional calculus as slope for
tangent-line.
In higher dimensions the coorinates for
$$df = \sum\limits_i {\frac{{\partial f}}{{\partial {x^i}}}(x)d{x^i}}$$
are the well known components for gradient, which lives in $({T_x}{R^n) }^ *$
in case of n-dim. vectorspace or manifold.
A: Your understanding of the calculus version of $dx$ is wrong.  While there are ways to do calculus using infinitesimals (non-standard analysis), $dx$ is certainly not the limit of $\Delta x$ as $\Delta x \to 0$: that would be simply $0$.  
One way to understand $dx$ and $dy$ in calculus is as new variables expressing changes in $x$ and $y$ on the tangent line to the graph of your function.  See e.g. Differential.
A: Intuitively they are same and can be taken as simple. I state about how it  appears to me. We consider small scalars and worry more about what is to be done among them in group operations rather than be bogged down what it deeply and definitively is.I say this as you are comfortable at advanced level and not so at the fundamental level.
The Pythagoras theorem $ s^2 = x^2 + y^2 $ in the plane has become in calculus in minuscule $ ds^2 = dx^2 +  dy^2 $. In differential geometry of two dimensions/ surface theory we have non-linear simple versions of the same curved coordinate lines $ ds^2  = E dx^2 + 2 F dx dy + G dy^2 $ which are now called metric, linked together in tangent space of a simple surface manifold. Further generalized into  multidimensional Riemannian manifolds.
