Calc 1.2a
- 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
- 11. Some reminders on how to deal with some special kinds of functions graphically: Piecewise graphs Look at #23 on page 55 Reminder of how to put piecewise functions in graphing calculator at http://www.tc3.edu/instruct/sbrown/ti83/funcpc.htm
- 12. http://www.prenhall.com/divisions/esm/app/graphing/ti83/Graphing/Special_graphs/greatest_integer/greatest_integer.html Greatest Integer graphs (stair steps) Look at #27 on page 56. Find “int” in your list of catalog items. Remember to use parentheses.
- 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
- 17. To summarize: For graphs that are continuous , to find limits, just evaluate f(c). For graphs with holes or jumps, look to see if the y-value approaches the same value L from both sides of x = c. If it does, regardless of what f(c) is, the limit exists and is L. For graphs that differ from right and left, are unbounded, or that oscillate as you approach x = c, the limit doesn’t exist.
- 18. Assignment 1.2a p. 54/ 1-21 odd, 25,27
Editor's Notes
- #3: Look at graph on document camera. Create table of values with table feature