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### Lesson 2-2

The Limit of a Function

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.

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

- Determine and Understand one-sided limits
- Determine and Understand two-sided limits

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

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

xa

lim f(x) = L

xa-

lim f(x) = L

xa+

lim f(x) = L

xa

lim f(x) = L

xa-

lim f(x) = L

xa+

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) = ∞

xa

lim f(x) = ∞

xa-

lim f(x) = ∞

xa+

lim f(x) = -∞

xa

lim f(x) = -∞

xa-

lim f(x) = -∞

xa+

x

Limits Using GraphsOne Sided Limits

Limit from right:

lim f(x) = 5

x10+

Limit from left:

lim f(x) = 3

x10-

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

lim f(x) = DNE

x10

Usually a reasonableguess would be:

lim f(x) = f(a)

xa

(this will be true forcontinuous functions)

ex: lim f(x) = 2

x2

but, lim f(x) = 7

x5

(not f(5) = 1)

and lim f(x) = DNE

x16

(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)

xa

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 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:
- 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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