do now – graph:. one-sided limits, sandwich theorem section 2.1b

Do Now – Graph:

1,0 1

1,1 2

2, 2

1,2 3

5,3 4

x x

x

f x x

x x

x x

One-Sided Limits, Sandwich TheoremSection 2.1b

One-Sided and Two-Sided LimitsSometimes the values of a function tend to different limitsas x approaches a number c from opposite sides…

Right-hand Limit – the limit of a function f as x approachesc from the right. lim

x cf x

Left-hand Limit – the limit of a function f as x approachesc from the left. lim

x cf x

–

+

One-Sided and Two-Sided LimitsWe sometimes call the two-sided limits of f

at c to distinguish it from the one-sided limits from the rightand left.

limx cf x

Theorem

lim limx c x cf x L f x L

–

A function f(x) has a limit as x approaches c if and only ifthe right-hand and left-hand limits at c exist and are equal.In symbols:

limx c

f x L

and

+

The Sandwich TheoremIf we cannot find a limit directly, we may be able to use thistheorem to find it indirectly…

If g x f x h x for all x cin some interval about c, and

lim lim ,x c x cg x h x L

then limx cf x L

The Sandwich TheoremGraphically………Sandwiching f between g and h forces thelimiting value of f to be between the limiting values of g and h:

y

x

L

c

f

h

g

Guided Practice

1

First, sketch a graph of the greatest integer function, thenfind each of the given limits. intf x x x

1

2

–1

–2

–1–2 2 3

2

lim intx

x

2

lim intx

x

2

lim intx

x

2

1

DNE

Guided PracticeSketch a graph of the given function, then evaluate limits forthe function at x = 0, 1, 2, 3, and 4.

1,0 1

1,1 2

2, 2

1,2 3

5,3 4

x x

x

f x x

x x

x x

Note: If f is not defined to the left of x = c, then f does nothave a left-hand limit at c. Similarly, if f is not defined to theright of x = c, then f does not have a right-hand limit at c.


0

limx

f x

At x = 0

1


1

limx

f x

At x = 1

0even though f(1) = 1

1

limx

f x

1Note: f has no limit as x 1 (why not???)


2

limx

f x

At x = 2

1

2

limx

f x

1

Note: even though f(2) = 2 2

lim 1x

f x


3

limx

f x

At x = 3

2

3

limx

f x

2

33 3

lim lim 2 3 limxx x

f x f x f f x


4

limx

f x

At x = 4

1

Note: At non-integer values of c between 0 and 4,the function has a limit as x c.

Guided PracticeFor the following, (a) draw the graph of f, (b) determine theleft- and right-hand limits at c, and (c) determine if the limitas x approaches c exists. Explain your reasoning.

2,c

3 , 2

2, 2

2, 2

x x

f x x

x x

Guided PracticeFor the following, (a) draw the graph of f, (b) determine theleft- and right-hand limits at c, and (c) determine if the limitas x approaches c exists. Explain your reasoning.

2

limx

f x

2

limx

f x

1

1

2

lim 1x

f x

Guided PracticeFor 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?

cos , 0

sec ,0

x xf x

x x

2x (0,1)

(a) , 2 2,

(b) c (c) c

do now – graph:. one-sided limits, sandwich theorem section 2.1b

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