Limits and Derivatives

[Jaltus-bot]

Can anyone give me a good, simple explanation of:

1. what limits are
2. what derivatives are
3. and how you get a derivative

Thank you

[Technomancer]

It's really pretty straightforward. Given some function f(x), the limit of f(x) as x approaches some point a, is the value that the function takes as you become arbitrarily close to the point a. For example:

lim e^(-x), as x-> +Inf is 0.

The derivitive of a function can be thought of as the slope of that function at a single point. In other words, you are computing the rate of change of the function with respect to the independent variable.

To compute the derivative, the basic formula is as follows:

f'(x)=lim (f(x+h)-f(x))/h as h->0

This means that you are computing the slope delta(f(x))/delta(x) for an infitessimaly small value of delta, which is the instantaneous rate of change of the function with respect to x.

http://mathworld.wolfram.com/Limit.html
http://mathworld.wolfram.com/Derivative.html

[Ingemar]

1. A limit describes the behaviour of a function f(x) as x approaches some number. For example:

lim x/x+1 (as x-->3) = 3/4.

(x--> is supposed to be a subscript under lim but such are the limitations of this board)

Notice how I just plugged in 3 for the value of x to get the answer. However, limits are not always cut and dry like that. The best way to determine a limit is to make a graph of your function (this is where a graphing calculator helps, but realize many math courses do not allow such a thing). Follow the curve along the x coordinate to the number that x approaches and the limit is the y value at that point in the curve.

This will really help when you have limits where x approaches infinity.

2. A derivative is the slope of a line tangent to your curve. (slope = difference in y/difference in x).

3. Look up a derivative chart.This works for particular types of functions.

[EricTheFred]

I won't repeat any of the math given above, as there's no point. But I will add a couple examples of why you need these things.

Limits:
You may have seen an exponential decay curve, which is where you have a function slowly dying away into a straight line. Mathematically, you can never get to the line, but it is pretty obvious that in the real world, that's where the curve is headed. That flat line that the curve is approaching, whether it is at y=zero or some other value, is "Lim x->infinity" (we pronounce it "the limit as x approaches infinity".)

Derivatives.
At some point, in Physics or Math, you have run into the kinematic equations which you use to calculate position, velocity and acceleration.
At the very least, you have learned to calculate that "hours time miles per hour equals miles traveled. " (d=v*t) for d=distance, v=velocity, t= time traveled.

The full equation is d=v0*t + (1/2)a*t^2 (v0 meaning initial velocity), which is for a constant acceleration (or if you can work out a function for it, for any acceleration).

and you possibly know that v=v0 + a*t .

v=v0+a*t is the Derivative of d=v0*t + (1/2)a*t^2
because velocity is the rate of change in distance, in other words, the rate at which you are changing distance.

If Velocity is constant, than distance versus time will graph as a straight line, because the slope of a straight line is a constant value. But if Velocity is changing (if you have an acceleration) than distance versus time will be curved or otherwise not a straight line, and the slope at different points is different. Derivitive calculation allows you to discover what that slope is at any given point, or create a function for it.