Do Now – Graph:

1. Do Now – Graph:

2. One-Sided Limits, Sandwich Theorem Section 2.1b

3. One-Sided and Two-Sided Limits Sometimes the values of a function tend to different limits as x approaches a number c from opposite sides… Right-hand Limit – the limit of a function f as xapproaches c from the right. + Left-hand Limit – the limit of a function f as xapproaches c from the left. –

4. One-Sided and Two-Sided Limits We sometimes call the two-sided limits of f at c to distinguish it from the one-sided limits from the right and left. Theorem A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols: + and –

5. The Sandwich Theorem If we cannot find a limit directly, we may be able to use this theorem to find it indirectly… If for all in some interval about c, and then

6. The Sandwich Theorem Graphically………Sandwiching f between g and h forces the limiting value of f to be between the limiting values of g and h: y h f L g x c

7. Guided Practice First, sketch a graph of the greatest integer function, then find each of the given limits. 2 1 –2 –1 1 2 3 –1 –2

8. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. Note: If f is not defined to the left of x = c, then f does not have a left-hand limit at c. Similarly, if f is not defined to the right of x = c, then f does not have a right-hand limit at c.

9. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 0

10. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 1 even though f(1) = 1 Note: f has no limit as x 1 (why not???)

11. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 2 Note: even though f(2) = 2

12. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 3

13. Guided Practice Sketch a graph of the given function, then evaluate limits for the function at x = 0, 1, 2, 3, and 4. At x = 4 Note: At non-integer values of c between 0 and 4, the function has a limit as x c.

14. Guided Practice For the following, (a) draw the graph of f, (b) determine the left- and right-hand limits at c, and (c) determine if the limit as x approaches cexists. Explain your reasoning.

15. Guided Practice For the following, (a) draw the graph of f, (b) determine the left- and right-hand limits at c, and (c) determine if the limit as x approaches cexists. Explain your reasoning.

16. Guided Practice For the following, draw the graph of f, and answer: (a) At what points c in the domain of f does limx  c exist? (b) At what points c does only the left-hand limit exist? (c) At what points c does only the right-hand limit exist? (0,1) (a) (b) (c)