That last nerve; Reaching your limit.
This is intended to be a "light and frothy" look at the idea of limits. There are many ways to perceive the concepts here in various degrees tedium, but I'm interested in helping you wrap your noggin around the concept!

Limits, something you reach when that high pitched, impossible to locate buzzing sound at your office just refuses to quit. In a light and frothy mathematical sense, a function's limit is "where the value appears to be going" when I put in some input value "x".
Let's look at some examples intuitively.
Let's look at some examples intuitively.
Let's say \( x=2 \). As I get really close to \( 2 \) on the \( x \)axis from either side, I see my \( y \) value also gets close to \(2\).

Let's say \( x=2 \) again. This time as I get close to \(2\) on the \(x\)axis from either side, my \(y\) value gets close to \(4\).

Surprise, let \(x=2\). However, we have a problem! As I get close to \( 2 \) on the \(x\)axis, \(y\) is \(1\) OR \(2\) depending on which side of \(x=2\) I'm on!

The last case there should have struck you as intuitively different from the rest. The left side of \( x=2 \) suggests the value is approaching \(1\), while the right side suggests \( 2 \). It would be nice if both sides agreed on where the value of the function is headed.
With this intuition we are ready to discuss 'one sided limits'.
With this intuition we are ready to discuss 'one sided limits'.
For a function \( f(x) \) to have a limit, we need both of the one sided limits to exist and be equal.
So what does it mean for a one sided limit to not exist? In short, it means that as I get close to \( x \) from my chosen side, the function value isn't tending toward anything. Possible, as in the case to the right, that there just isn't a function value there.
Check out the graph on the right for \( \lim{x_0^} f(x) \). Here, as I come at zero from the left side, there isn't any graph! (You'll have to trust that I'm not cropping graph out in some dastardly sneaky trick, I wouldn't do that to you). Sometimes though, something even stranger happens. 
This terrible creature, \( f(x) = \sin{( \frac{1}{4x} )} \), is an example of a one sided limit failing to exist due to extreme oscillation.
As \( x \) gets closer and closer to zero from the right, the input of the sine function gets larger and larger, exploding to infinity, and causing the sine wave to oscillate so frequently it's impossible to hone in on a 'limiting value'. So you see, there are a couple of ways a one sided limit can fail to exist. Now all we need to ensure is that they are equal! 
We've already seen cases where both one sided limits agree or disagree. (Remember, if one or both fails to exist, the game is already over, so we are assuming now that they both exist.)
If both one sided limits exist and agree, we say that \( \lim_{x \to x_0} f(x) \) exists and is equal to that agreed upon value. Notice in that expression that there isn't an indicator about which 'side' we're approaching from. This is the function's actual limit at \( x_0 \).
Don't worry if you're still confused.
This lesson is only the first of several. The concept of limit is a rather large one, even a "light and frothy" take. We still need to know how to evaluate limits without a graph, consider what happens if \( x \) gets infinitely positive or negative, among other things. It will all become clear, grasshopper. 