If my acceleration is -1 ($a=-1\:\rm{m/s^2}$) and I'm standing in the infinite ($x_0=\infty \:\rm m$), could I reach the point $x=0\:\rm m$? I'm standing in the infinite where $x_0=\infty \:\rm m$. If I have a negative acceleration, could I reach the point $x=0\:\rm m$? Would it be possible to calculate how long would take to reach the point $x=0\:\rm m$?
 A: (Mathematicians, please hold back the pitchforks--i'll update this with a mathematically correct version later.)
$$s=ut+\frac{1}{2}at^2$$,$$\text{where }u=0,a=-1,s=-\infty$$
$$\therefore-\infty=0+\frac{1}{2}(-1)t^2$$
$$\therefore t=\sqrt{2\infty}=\infty \:^\text{  {*}}$$
So you will never reach $s=0$. Of course, standing at $s=\infty$ is impossible anyways. In physics, infinity is a place sufficiently removed from a system to be free of all influences from it. For example, a hundred meters is infinity for a system consisting of a marble, where you only consider gravity. So it's a relative concept. In mathematics, infinity is something that you can try to reach but never do--it's not a real number(in both the lay and technical meanings of "real number").
A more mathematically correct way to do it would be to use limits and show that $t$ diverges as $s$ approaches $-\infty$.
*Note that even though I am using it like a normal number, $\infty$ is not a number and arithmetic does not work on it. $\infty+\infty=\infty\times\infty=\infty^\infty=\infty$ is OK to use, but $\infty-\infty,\infty\times0,\infty/\infty$ are all undefined like $0/0$ and should not be used.
A: It's same as you start from $x_0=0\:\rm m$ and have $a=1\:\rm{m/s^2}$. You will aim to $\infty$, but never reach it.
Sorry, didn't find how to write with math code.
A: The given answers are not really correct. Manishearth should have followed his intuition. Maybe this might help:
$$ x_0 = \infty \\x(t) = x_0 +\frac{1}{2}at^2$$
Now can we reach a certain point after waiting for a long time? $$\lim_{t\rightarrow \infty}x(t) = x_0 +\frac{1}{2}at^2= x_0 - \infty \\ \lim_{t\rightarrow \infty}x(t) = \infty - \infty$$
So now where are stuck and ask our math guys what is $\infty - \infty$? and it turns out that we just forgot how we came to our starting position. If we do not know that (i.e. which limit brought us there), then the answer is not defined as just the symbol $\infty$ has no memory. So yes, we could certainly be at $0$, but also at $-\infty$ or$\infty$. A precise answer can only be given if we know how we got there. 
This is also true for the time. If you do not know how far you are out you cannot calculate how long it will take to come back. 
