What is the difference between a limit and a derivative?

I had previously thought they were the same thing, but certain problems, like the following make me think twice:

f(x) = x^2 - 5x - 2

For the above function, if I look for the limit using an intuitive method, such as looking at what y-value the function nears as x approaches a number (say x -> 3, for instance), I get ( -8). However, when I calculate the slope at x = 3, (using the average of secants left and right of (3, -8)), I get a slope of 1. If I check the derivative at 3,-8 for this function, I also get 1. If I take the limit in Mathematica, I get -8 (which agrees with my lazy, intuitive approach from the start).

So, it looks like the limit as x approaches 3 for f(x) = x^2 - 5x - 2 is -8, but the derivative is 1. These are two very different numbers, so it seems that limits and derivatives are also different concepts. What is the difference?