Limits vs. Derivatives

[kamikaze]

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?

[stapel_eliz]

How are you taking the limit?

[kamikaze]

How are you taking the limit?

I am looking at the values of y as x approaches 3 using the table function on a Ti-83 graphing calculator.

From the left and right, the function seems to approach -8 at a uniform rate around x=3. Based on what I've been taught so far about limits, that should make the limit -8.

Also, my "understanding" prior to this had been that:

limit = slope of a tangent line = rate of change = derivative

UPDATE:

I think I've got it:

Not all limits are derivatives, but all derivatives are limits.

In other words, the derivative is a specific kind of limit (using, in one form, the difference quotient).

If I've got this right, then I sure wish textbook authors would see the value of pointing stuff like this out to thick-headed folks like myself.

[stapel_eliz]

If you're taking the limit of y as x approaches 3, then you're doing the following:

. . . . .

Since y is a polynomial, then the limit is the evaluated value:

. . . . .

But the function is not (usually) equal to its derivative. Instead, try applying the derivative-as-a-limit formula they gave you, which should be something like:

. . . . .

See where that leads.