1. This document discusses finding limits graphically and numerically using tables of values and graphs. It explains how to determine if a limit exists or does not exist based on the behavior of the function as it approaches the given value. 2. Limits can exist even if the function is not defined at a point. The limit depends on whether the function values approach a single number from both sides, not on the actual function value at that point. 3. There are special types of functions like piecewise functions and greatest integer functions that require specific approaches to graph. Limits can fail to exist if the function behavior differs on each side, becomes unbounded, or oscillates without approaching a single value.

1. 1.2 Finding Limits Graphically and Numerically What do you do when infinity approaches? TAKE IT TO THE LIMIT!

2. Let’s sketch , x ≠ 1 There is a hole at x = 1 To get an idea of what is happening at x = 1 you could use values that approach 1 from the left, and values that approach 1 from the right. p. 48 figure 1.5

3. Put graph in calculator, go to tblset and select Indpnt: Ask and Depend: Auto. Then select table and input values getting closer to x = 1 from both directions

4. Although x cannot equal 1, you can get arbitrarily close to 1, and then f(x) moves arbitrarily close to 3. Using limit notation, This is read as “the limit of f(x) as x approaches 1 is 3”

5. Limit definition informally: If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c , is L, written as: You can find a limit numerically by looking at table of values, graphically by graphing and zooming in, and analytically, which is Sec 1.3.

6. A lot of functions are continuous and have no holes or jumps. To find limits, you just put in the x = c that you are approaching, and out pops your limit. But how interesting are those graphs? We want to look at graphs that have something unusual happening at x = c and find out the limit in that case!

7. How are these graphs alike and different? What conclusions can you make about what a limit is?

8. The existence or non-existence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x aproaches c! In other words, what f(c) becomes is NOT ALWAYS the limit!

9. Ex. 1, p. 49: Estimating a limit numerically Use a table and choose values near x = 0 to estimate What happens if we try direct evaluation of f(0)?

10. Numerical approach Graphical approach

13. Limits that Fail to Exist Ex 3, p. 50: Behavior that Differs from the Right and the Left Show that the limit does not exist for For positive x-values, For negative x-values, Since there is not just one value it approaches, it does not exist.

14. Limits that Fail to Exist Ex 4, p. 50: Unbounded Behavior Show that the limit does not exist for As x approaches zero from either the left or right, f(x) increases without bound. By choosing an x close enough to zero, you can make f(x) as large as you want. Since f(x) is not approaching a real number, the limit does not exist.

15. Limits that Fail to Exist Ex 5 p.51: Oscillating Behavior Look at -1.2 -0.25 Notice that in the window the graph starts to oscillate between -1 and 1. You can’t always trust your graph. The limit fails to exist

18. Assignment 1.2a p. 54/ 1-21 odd, 25,27