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 x approaches
c from the right. +
Left-hand Limit – the limit of a function f as x approaches
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 for all in some interval about c, and then
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 y h f L g x c
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 At x = 1 even though f(1) = 1
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 At x = 2 Note: even though f(2) = 2
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 c exists. 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 c exists. 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 lim x  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)