What application do they serve. Because I am pretty sure we cannot say that we have a number going towards an object from both sides.
Take a piece of string and chop it in half. Arrange both pieces so that one end of one piece almost touches on end of the other. Now follow each string from the other end to the point where they almost touch. This is the limit which you have reached from both ends.
Limits come into play whenever there is a physical property being measured that changes over time. If you sample the value of that property on a regular basis you get a good approximation of the rate of change of that property, but it's only an approximation. The true rate of change requires measuring the property as the time interval between measurements approaches zero. For example, to measure the velocity of an object at a certain point in time you could measure the object's displacement just before that time and again just after, calculate the distance traveled, and then divide by the time interval. Suppose you had an object that is driven by some force such that its displacement is x = t^3. Measurements of its displacement would yield the following results:
Time(s), Distance(m)
0, 0
1, 1
2, 8
3, 27
What is its velocity at t=2 seconds? Comparing displacements at t= 1 and t=3 yields (27-1)/2 = 13 m/s. But now suppose you were able to measure displacements 4 times per second instead of once per second - you would get the following data:
Time, Distance
1.75, 5.36
2, 8
2.25, 11.39
and the calculated velocity at t=2 is (11.39-5.36)/0.5 = 12.06 m/s. This is a more accurate measurement, but still not perfect. To get the "true" velocity at t=2 requires finding the limit of the change in displacement as the time interval approaches zero:
$\displaystyle v(t=2s) = \displaystyle \lim_{\Delta t \to 0} \frac {\Delta x}{\Delta t} $
For anyone who has studied calculus this clearly the derivative of x(t) evaluated at t= 2 seconds, which yields a "true" answer of 12 m/s. In real life there is no way to take instantaneous measurements like this, so the quest for accuracy means trying to get the time interval to as short a period as possible (i.e. try to drive delta t to 0).
One can do "discrete mathematics" without limits (that's pretty much the definition of "discrete mathematics"!) but any time you want to work with "continuous" or "instantaneous change" (as opposed to "average change") you need the limit.