Sunday, October 2, 2011

Lesson 1.1: The Concept of Limits

Let x0 be in the domain of the function ƒ(x) and let y = ƒ(x). The limit of ƒ(x) means the value that ƒ(x) approaches, or gets closer and closer to (arbitrarily close to), as x approaches x0. If y0 is the limit of ƒ(x) as x approaches x0, this can be written as:

As a simple example, take the function ƒ(x) = x2 and let x0 = 2. In this case, limx→2 (x2) = 4. What this means that as x approaches 2, ƒ(x) gets arbitrarily close to the value 4. To see what is happening here, imagine the graph of this function (which is a parabola opened up and passing through the origin):



If you take your finger and trace the curve in that graph, as you finger gets closer and closer to 2 along the x axis, it gets closer and closer to 4 along the y axis.

In this particular case,ƒ(x0) = limxx0 ƒ(x) (i.e., limx→2 (x2) = 4 = 22). This is not always the case.

To see an example where this is not true, consider the function ƒ1(x) = (x4 - 1)/(x - 1). This function is not defined for x = 1 (at x = 1, the denominator is zero). You can see this in the graph below, where a hole is in the plot of the function at x = 1:



Because this function is undefined at x = 1, in order to find the limit as x approaches 1, it is useful to calculate the function for values of x close to 1 on both sides and see if the numbers are trending to a specific value:


x
0.75
0.9
0.99
0.999
1
1.001
1.01
1.1
1.25
ƒ1(x)
2.734
3.439
3.940
3.994
undefined
4.006
4.060
4.641
5.766

As you can guess from the table above, as x approaches 1 from either side, ƒ1(x) gets arbitrarily close to 4. Thus,



A bit of algebraic manipulation reveals that the numerator of this function can be factored as into (x3 + x2 + x + 1)(x - 1). If we cancel x - 1 in the numerator with x - 1 in the denominator we get the function ƒ2(x) = x3 + x2 + x + 1. This function is identical to ƒ1(x) except at the point x = 1, as you can see in the graph below:


As you have no doubt noticed, ƒ2(1) = 4, and as you may have guessed, it is not a coincidence that limx→1 ƒ1(x) = ƒ2(1). We shall see the reason for this shortly.

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