SECTION 2.2Finding Limits Graphically & Numerically

What’s the point of Calculus?

What’s the point of Calculus?

What’s the point of Calculus?

What’s the point of Calculus?

The Concept of a “Limit”

lim𝑥→𝑐

𝑓 (𝑥 )=L

Example 1Use a table of values to estimate the limit. Confirm graphically.

-5.1 -5.01 -5.001 -5 -4.999 -4.99 -4.9

Example 2(#18-26 even in your book)

18.

20.

Example 2 (cont.)(#18-26 even in your book)

22.

24.

Example 2 (cont.)(#18-26 even in your book)

26.

Example 3(#28 in your book)

a.

b.

c.

d.

e.

f.

g.

h.

(#30 in your book)

Use the graph of to identify the values of for which exists.

Example 4

Example 5Sketch a graph of a function with the given properties.

DNE

SECTION 2.3Evaluating Limits Analytically

Theorems Involving Limits

Theorem 2.1 Some Basic Limits (p. 79)

Let and be real numbers and let be a positive integer.

1. 2. 3.

Theorems Involving Limits

Theorem 2.2 Properties of Limits (p. 79)

Let and be real numbers, let be a positive integer, and let and be functions.

1. Scalar Multiple:

2. Sum or difference:

3. Product:

4. Quotient: , provided

5. Power:

Theorems Involving Limits (cont.)

Theorem 2.3 Limits of Polynomial and Rational Functions (p. 80)

If is a polynomial and is a real number, then

.

If is a rational function given by and is a real number such that , then

.

Theorems Involving LimitsTheorem 2.4 The Limit of a Function Involving a Radical(p. 80)

Let be a positive integer. The following limit is valid for all if is odd, and is valid for if is even.

Theorem 2.5 The Limit of a Composite Function (p. 81)

Example 1Find .

Example 2Find .

Example 3Find .

Example 4Find .

Other Theorems Involving Limits

• Theorem 2.6 deals with finding the limits of trigonometric, exponential, and logarithmic functions.

• Theorem 2.7 talks about fnc.’s that agree at all but one point.

• Theorem 2.8 is the Squeeze Theorem.

Example 5Find .

Theorem 2.9

1. 2. 3. Find

Example 5

Find given .

Limits of Transcendental Functions

Example 6Find the limit if it exists.

a.

b.

Functions Agreeing at All But One Point

Example 7Find .