2. Self-Similarity When we zoom in 200% on the center of the concentric circles, the figure we see looks exactly like the original figure. In other words, the new piece of the figure we are looking at is similar to the original figure. Self-similarity is an important property of figures known as…

3. Fractals Fractals are figures that can be split into multiple new figures, each of which is a reduced-sized copy of (i.e, self-similar to) the original figure. Fractals can usually be defined very easily mathematically (but we won’t go into that today). Let’s look at an example.

6. Sierpiński Triangle We started with a triangle with area 1. After each iteration:  We have 3 times as many triangles, and the area of each triangle is ¼ that of the triangles in the previous iteration.  Thus, each iteration yields an area that is ¾ that of the previous iteration. What happens if we keep going?

13. Sierpiński Triangle We will eventually end up with an infinite number of triangles, each with an infinitely small area. Essentially, the amount of black space in the triangle gets closer and closer to zero… But is that possible? Can it ever actually reach zero?

14. What’s Going On? After the nth iteration, the area of the figure is given by the function A(n)=(¾) n. As n gets larger, the value of the function gets smaller and smaller… But we can’t say n equals infinity, because infinity is not a number! Thus, as n approaches infinity, A tends to 0, but technically never reaches it.

15. The Limit This concept of approaching a value without necessarily reaching it is known as the limit, and is the foundation for everything in calculus. The way we can express our area function for the Sierpiński Triangle is...

16. The Limit “The limit as n approaches infinity of the area function is zero.” lim A(n) = 0 n→∞

17. The Limit: A Graphical Approach Fractals are a neat application which we can explore in more depth later on in the year. In the meantime, let’s take a look at limits in the context of something a little more familiar: graphs of functions.

20. The Limit: A Graphical Approach A limit is a value we approach, but do not necessarily reach. A limit is a value which tells us about the “journey” of the function as the input variable approaches a certain value, but not necessarily the final “destination” of the function when the input variable equals that value.

22. One-Sided and Two-Sided Limits Left-hand limit:Right-hand limit: The two-sided limit is undefined, because the two one-sided limits are different: lim f(x) = –1 x→2 – lim f(x) = 1 x→2 + lim f(x) = Ø x→2

23. One-Sided and Two-Sided Limits A two-sided limit exists if and only if the two one-sided limits equal one another. If the two one-sided limits both equal some value L, then the two-sided limit also equals L (and vice versa). If the two one-sided limits equal two different values, then we say the (two- sided) limit does not exist, is undefined, or (symbolically) = Ø.

24. “The limit does not exist!” “Mean Girls” TM & Copyright © 2004 Paramount Pictures.