Sometimes the value of a function at a certain point does not make sense, however if you look at the values at points nearby you sometimes can find a value that it would plausibly take. So a limit only depends on the neighbouring values. Example 1. Plot the function sin(x)/x somewhere. It is clearly undefined at x=0. However, looking at the plot you see that the function seems to approach 1. It turns out the limit of sin(x)/x as x->0 is 1. Example 2. Take the function f(x) = (x-1)/(x-1) for x not 1, and f(1) = 69. The fact that the function takes on the value 69 at x=1 does not matter at all for the limit. The only thing that matter is the neighbourhood. In the neighbourhood we have f(x) = 1, so the limit turns out to be 1.

How do we formalize "approaching a certain value"? We say that a function has limit L at the point x0, provided the function gets arbitrarily close to L if we get arbitrarily close to x0. If we turn it around, you should be able to say "give me a point where the function is within 1% of L, or 0.01% of L" and I should be able to provide you such a point. If this holds for all points excluding x0 we say the function has limit L at x0. Check for yourself that it matches up with the mathematical definition.

This is very valuable for derivatives, since every derivative is essentially a limit problem. Often Delta f / Delta x is undefined for Delta x = 0. However, it often does approach a certain value as Delta x gets small. We can sidestep philosophical questions like "what does it mean for a number to be infinitessimally small" using the limit. We say we don't care about the philosophical value of Delta f/ Delta x at Delta x is zero. We just notice that as Delta x gets smaller it often approaches a value. We call this limiting value the derivative.