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Next, we turn to the issue of exactly which objects are sets and which objects are not. The situation is analogous to how we defined the natural numbers in the previous chapter; we started with a single natural number, 0, and started building more numbers out of 0 using the increment operation. We will try something similar here, starting with a single set, the empty set, and building more sets out of the empty set by various operations. We begin by postulating the existence of the empty set.

But the second limit is divergent (because x- 3 goes to infinity as x --+ 0, and cos(x- 4) 2x sin(x- 4 )-4x- 2 cos(x- 4 ) . . does not go to zero ) . So the hm1t hmx--+0 1 diverges. One might then conclude using L'Hopital's rule that 2 . ( -4) limx--+0 _x smx x also diverges; however we can clearly rewrite this limit as limx--+0 xsin(x- 4), which goes to zero when x--+ 0 by the squeeze test again. 5), but it still requires some care when applied. 13 (Limits and lengths). When you learn about integration and how it relates to the area under a curve, you were probably presented with some picture in which the area under the curve was approximated by a bunch of rectangles, whose area was given by a Riemann sum, and then one somehow "took limits" to replace that Riemann sum with an integral, which then presumably matched the actual area under the curve.

2). As the above examples show, we can now create quite a few sets; however, the sets we make are still fairly small (each set that we can build consists of no more than two elements, so far). The next axiom allows us to build somewhat larger sets than before. 4 (Pairwise union). Given any two sets A, B, there exists a set A U B, called the union A U B of A and B, whose elements consists of all the elements which belong to A or B or both. In other words, for any object x, x E AUB {:::=> (x E A or x E B).

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