If the values of a function can be made as close as we ...



I. Introduction – Limits Review:

A. Graph each function and find the limit as x approaches 1.

[pic] [pic] [pic]

|[pic] |[pic] |[pic] |

B. Let [pic]

(1). Find [pic]

(2). Sketch a graph of f and illustrate the limit in part (1) graphically

[pic]

C. Let [pic]

(1). Find [pic]

(2). Sketch a graph of g and illustrate the limit in part (1) graphically

[pic]

II. One-sided limits

Informal notion of a limit from the left: If the values of a function [pic]can be made as close as we like to L by taking values of x sufficiently close to a (where [pic]) then we write [pic] which is read “the limit of f(x) as x approaches a from the left is L.

Informal notion of a limit from the right: If the values of a function [pic]can be made as close as we like to L by taking values of x sufficiently close to a (where [pic]) then we write [pic] which is read “the limit of f(x) as x approaches a from the right is L.

Theorem 3: [pic] if and only if [pic]=[pic]

Applications of Theorem 3:

1. Lend graphical and numerical support for the claim that [pic]where[pic].

2. Lend graphical and numerical support for the claim that [pic] DNE where [pic].

3. Lend graphical and numerical support for the claim that [pic] DNE where [pic].

4. Sketch [pic] and find, if possible …

(a) [pic]

(b) [pic]

(c) [pic]

5. Sketch [pic] and find, is possible …

(a) [pic]

(b) [pic]

(c) [pic]

6. Sketch [pic] and find, if possible …

(a) [pic]

(b) [pic]

(c) [pic]

III. Definition of a Limit

DEFN1: Definition of a Limit of a Function:

Let a function f be defined on an open interval containing a, except possibly at a itself, and let L be a real number. The statement [pic] means that for every [pic], there is a [pic] such that if [pic], then [pic].

➢ We sometimes call the inequality [pic] a delta tolerance statement and the inequality [pic] an epsilon tolerance statement.

DEFN2: Alternate definition of Limit:

[pic] means that for every [pic], there is a [pic] such that if x is in the open interval [pic]and [pic], then f(x) is in the open interval [pic].

1. Use DEFN1 to prove that [pic] for [pic]

2. Use DEFN1 to prove that [pic] for [pic]

3. Use DEFN1 to prove that [pic]

IV. Properties of Limits

Theorem 1.0: Properties of Limits

a*. [pic], for some constant c. Prove using the Definition of a Limit

b*. [pic] Prove using the Definition of a Limit

Theorem 1.1: Properties of Limits

If [pic]and [pic]both exist, then

a. [pic]

b. [pic]

c. [pic]

d. [pic], provided [pic]

e. [pic], for some constant c.

Each property can be proved using the Definition of a Limit

Theorem 1.2: Property of Limits

a*. [pic] Prove using Theorem 1.1

b*. [pic] Prove using Theorem 1.1

Theorem 2.0: Polynomial and Rational Functions

a*. If f is a polynomial function and a is a real number, then [pic]

Prove using Theorem 1.2

b*. If q is a rational function and a is in the domain of q, then [pic]

Prove using Theorem 1.2

Theorem 1.3: Properties of Limits

a*. If a > 0 and n is a positive integer, or if [pic]and n is an odd positive integer, then

[pic] Prove using Theorems 1.2 and 2.0

b. If a function f has a limit as x approaches a, then

[pic], provided either n is an odd positive integer or n is an even positive integer and [pic]

Theorem 4: The Sandwich Theorem

Suppose [pic] for every x in the open interval containing a, except possible at a.

If [pic], then [pic]

Example: Prove [pic]

V. Functions Increasing or Decreasing Without Bound

If the value of a function [pic]increases without bound as x approaches a number a from the left, we say [pic]

If the value of a function [pic]increases without bound as x approaches a number a from the right, we say [pic]

If the value of a function [pic]decrease without bound as x approaches a number a from the left, we say [pic]

If the value of a function [pic]decreases without bound as x approaches a number a from the right, we say [pic]

In all cases above, the notation is somewhat misleading. None of the limits above are actually defined. Using the equal symbol is misleading because the limit does not actually equal anything. We use [pic]as an indication of which direction the function is diverging.

Example: Find [pic]

Example: Find [pic]

Definition: The line [pic]is a vertical asymptote of the graph of a function [pic]if [pic]or [pic]

VI. What happens to a function when x increases or decreases without bound?

Theorem[pic] where c is a constant

Theorem [pic] where c is a constant

Example: Find [pic]

Example: Find [pic]

Example: Find [pic]

Definition: The line [pic]is a horizontal asymptote of the graph of a function [pic]if [pic].

Final note: The expression [pic]is actually the one-sided limit:

VII. Continuous Functions

Definition: A function f is continuous at a number c if [pic]

When proving that a function f is continuous at a number c, show the following conditions are satisfied:

i) f(c) is defined

ii) [pic]exists

iii) [pic]

A function f is not continuous at a number c if [pic]

This inequality will occur if any of the following are true …

i) f(c) is not defined … because then f(c) could not equal the limit

ii) [pic]fails to exist or [pic]fails to exist … because then [pic]fails to exist and cannot equal f(c)

iii) [pic]… because then [pic]fails to exist and cannot equal f(c)

Types of Discontinuities: Removable (hole), Infinite, Jump

Let [pic]. Graph f. Prove that f is not continuous at [pic]. Define f such that it is continuous at [pic].

[pic]

Let [pic]. Prove that g is or is not continuous at [pic]

Let [pic]. Prove that h is or is not continuous at [pic].

Theorem: Continuous Polynomial Functions

i) A polynomial function f is continuous at every real number c.

ii) A rational function [pic]is continuous at every number except the numbers c such that [pic]

Definition: A function f is continuous at a left endpoint a if [pic]

Definition: A function f is continuous at a right endpoint b if [pic]

Definition: A function f is continuous on an interval if it is continuous at every point on the interval.

Definition: A continuous function is one that is continuous at every point if its domain.

Example: Prove that [pic]is continuous on the close interval [-3, 3]

Theorem 6: Properties of Continuous Functions

Theorem 7.0: If [pic]and if g is continuous at b, then [pic]

Theorem 7.1: Composite of Continuous Functions

If f is continuous at c and g is continuous at f(c), then the composite function [pic]is continuous at c;

that is, [pic]

Example: If [pic], show that k is continuous at every real number.

VIII. Intermediate Value Theorem

Theorem 8: The Intermediate Value Theorem for Continuous Functions

A function [pic]that is continuous on a closed interval [a, b] takes on every value between f(a) and f(b).

Example: Let [pic] with corresponding table of values:

|x |-4 |-3 |-2 |-1 |0 |1 |2 |

|f(x) |-139 |72 |41 |2 |-3 |-4 |17 |

Use the intermediate value theorem to show that f has three zeros in the interval [-4, 2]

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