Evaluating Limits Analytically

Lesson 2.3

2

What Is the Squeeze Theorem?

Today we look at various properties of limits, including the Squeeze Theorem

Today we look at various properties of limits, including the Squeeze Theorem

3

Basic Properties and Rules

Constant rule

Limit of x rule

Scalar multiple rule

Sum rule(the limit of a sum is the sum of the limits)

limx ck k

limx cx c

lim ( ) lim ( )x c x c

k f x k f x

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

See other properties pg. 79-81

See other properties pg. 79-81

4

Limits of Functions

Limit of a polynomial P(x)• Can be demonstrated

using the basic properties and rules

Similarly, note the limit of a rational function

lim ( ) ( )x cP x P c

( )Given ( )

( )

( )lim ( )

( )x c

P xQ x

D x

P cQ x

D c

What stipulation must be made concerning

D(x)?

What stipulation must be made concerning

D(x)?

5

Try It Out

Evaluate the limits• Justify steps using properties

3 2

0lim 5 4x

x x

1

0

sinlim

1x

x

x

6

General Strategies

7

Some Examples

Consider

• Why is this difficult?

Strategy: simplify the algebraic fraction

2

2

6lim

2x

x x

x

2

2 2

2 36lim lim

2 2x x

x xx x

x x

8

Reinforce Your Conclusion

Graph the Function• Trace value close to

specified point

Use a table to evaluateclose to the point inquestion

9

Some Examples

Rationalize the numerator of rational expression with radicals

Note possibilities for piecewise defined functions

4

2lim

4x

x

x

2

2

3 2 2( )

5 2

lim ( ) ?x

x if xf x

x if x

f x

10

Three Special Limits

Try it out!

0

sin 4lim ?

9x

x

x 20

1 coslimx

x

x

1

0 0 0

sin 1 coslim 1 lim 0 lim 1 xx x x

x xx e

x x

View GraphView

GraphView

GraphView

GraphView

GraphView

Graph

11

Squeeze Rule

Given g(x) ≤ f(x) ≤ h(x) on an open interval containing cAnd …

• Then

lim ( ) lim ( )

lim ( )

x c x c

x c

g x h x L

f x L

12

Assignment

Lesson 2.3A Page 87 Exercises 1-43 odd

Lesson 2.3B Page 88 Exercises 45 – 97 EOO