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Lesson 2-2. The Limit of a Function. Transparency 1-1. y. x. k. (6,4). C. (0,1). (-6,-2). A. B. 5-Minute Check on Algebra. 6x + 45 = 18 – 3x x 2 – 45 = 4 (3x + 4) + (4x – 7) = 11 (4x – 10) + (6x +30) = 180 Find the slope of the line k.

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Lesson 2 2

Lesson 2-2

The Limit of a Function


Transparency 1-1

y

x

k

(6,4)

C

(0,1)

(-6,-2)

A

B

5-Minute Check on Algebra

  • 6x + 45 = 18 – 3x

  • x2 – 45 = 4

  • (3x + 4) + (4x – 7) = 11

  • (4x –10) + (6x +30) = 180

  • Find the slope of the line k.

  • Find the slope of a perpendicular line to k

Standardized Test Practice:

A

B

C

D

-2

1/2

2

-1/2

Click the mouse button or press the Space Bar to display the answers.


Transparency 1-1

y

x

k

(6,4)

C

(0,1)

(-6,-2)

A

B

5-Minute Check on Algebra

  • 6x + 45 = 18 – 3x

  • x2 – 45 = 4

  • (3x + 4) + (4x – 7) = 11

  • (4x –10) + (6x +30) = 180

  • Find the slope of the line k.

  • Find the slope of a perpendicular line to k

9x +45 = 18 9x = -27 x = -3

x² = 49 x = √49 x = +/- 7

7x - 3 = 11 7x = 14 x = 2

10x + 20 = 180 10x = 160 x = 16

∆y y2 – y1 4 – 1 3 1

m = ----- = ----------- = -------- = ------ = ----

∆x x2 – x1 6 – 0 6 2

∆y

∆x

Standardized Test Practice:

A

B

C

D

-2

1/2

2

-1/2

Click the mouse button or press the Space Bar to display the answers.


Objectives
Objectives

  • Determine and Understand one-sided limits

  • Determine and Understand two-sided limits


Vocabulary
Vocabulary

  • Limit (two sided) – as x approaches a value a, f(x) approaches a value L

  • Left-hand (side) Limit – as x approaches a value a from the negative side, f(x) approaches a value L

  • Right-hand (side) Limit – as x approaches a value a from the positive side, f(x) approaches a value L

  • DNE – does not exist (either a limit increase/decreases without bound or the two one-sided limits are not equal)

  • Infinity – increases (+∞) without bound or decreases (-∞) without bound [NOT a number!!]

  • Vertical Asymptote – at x = a because a limit as x approaches a either increases or decreases without bound



Limits
Limits

When we look at the limit below, we examine the f(x) values as x gets very close to a:

read: the limit of f(x), as x approaches a, equals L

One-Sided Limits:

Left-hand limit (as x approaches a from the left side – smaller)

RIght-hand limit (as x approaches a from the right side – larger)

The two-sided limit (first one shown) = L if and only if both one-sided limits = L

if and only if and

lim f(x) = L

xa

lim f(x) = L

xa-

lim f(x) = L

xa+

lim f(x) = L

xa

lim f(x) = L

xa-

lim f(x) = L

xa+


Vertical asymptotes
Vertical Asymptotes

  • The line x = a is called a vertical asymptote of y = f(x) if at least one of the following is true:

lim f(x) = ∞

xa

lim f(x) = ∞

xa-

lim f(x) = ∞

xa+

lim f(x) = -∞

xa

lim f(x) = -∞

xa-

lim f(x) = -∞

xa+


Limits using graphs

y

x

Limits Using Graphs

One Sided Limits

Limit from right:

lim f(x) = 5

x10+

Limit from left:

lim f(x) = 3

x10-

Since the two one-sided limits are not equal, then

lim f(x) = DNE

x10

Usually a reasonableguess would be:

lim f(x) = f(a)

xa

(this will be true forcontinuous functions)

ex: lim f(x) = 2

x2

but, lim f(x) = 7

x5

(not f(5) = 1)

and lim f(x) = DNE

x16

(DNE = does not exist)

2

5

10

15

When we look at the limit below, we examine the f(x) values as x gets very close to a:

lim f(x)

xa


Example 1
Example 1

  • Answer each using the graph to the right (from Study Guide that accompanies Single Variable Calculus by Stewart)

Lim f(x) =

x→ -5

4

Lim f(x) =

x→ 2

3

DNE

Lim f(x) =

x→ 0

Lim f(x) =

x→ 4

0


Example 2
Example 2

sin x

Lim ------------ =

x

Use tables to estimate

x→ 0


Example 3
Example 3

Use algebra to find:

a.

b.

c.

x³ - 1

Lim ------------ =

x - 1

Lim (x² + x + 1) = 3

x→ 1

x→ 1

x - 1

Lim ------------ =

x - 1

Lim (x + 1) = 2

x→ 1

x→ 1

x 1

Lim --------- - -------- =

x – 1 x – 1

Lim 1 = 1

x→ 1

x→ 1


Summary homework
Summary & Homework

  • Summary:

    • Try to find the limit via direct substitution

    • Use algebra to simplify into useable form

  • Homework: pg 102-104: 5, 6, 7, 9;


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