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Quiz #2 Review

Unit 2. Limits and Derivatives The Limit of a Function, Limit Laws,

The Definition of a Limit, Continuity,

and The Intermediate Value Theorem

Suggested Problems

Evaluate the limits of functions.

Evaluate the limits of functions using the graph on left.

Suggested Problems - SOLUTIONS -

Evaluate the limits of functions.

1 0

-9 0

Evaluate the limits of functions using the graph on left.

1 3

0 About 1

DNE About 1

0 About 1

Existence of Limit

Evaluate following limits using the graph.

if and only if .

Infinite Limits and Vertical Asymptotes

Using the given graph of answer the questions.

The line is called a vertical asymptote of the curve , if at least one of the following infinite limits is true:

Evaluate following limits. Identify vertical asymptotes, if exist.

Group Discussion

Using a calculator, fill in the table below for the function

Using the values above, evaluate following limits.

Find vertical asymptotes, if exist.

x 0.9 0.99 0.999 1.001 1.01 1.1

f(x)

x -0.9 -0.99 -0.999 -1.001 -1.01 -1.1

f(x)

Suggested Problem

Using a calculator, fill in the table below for the function

Using the values above, evaluate following limits.

Find vertical asymptotes, if exist.

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x)

x -0.9 -0.99 -0.999 -1.001 -1.01 -1.1

f(x)

Suggested Problem - SOLUTIONS -

Using a calculator, fill in the table below for the function

Using the values above, evaluate following limits.

-3 -3 -3

0 0 0

Find vertical asymptotes, if exist. None.

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x) -0.1 -0.01

x -0.9 -0.99 -0.999 -1.001 -1.01 -1.1

f(x)

Suggested Problems

Textbook. p107

12) 14)

49) c) Graph g(x)

a) Find and

b) Does exist?

Suggested Problems - SOLUTIONS -

Textbook. p107

12) 14)

3/5 D.N.E

49) c) Graph g(x)

a) Find and

b) Does exist? No

Geometrical Interpretation of the Limit Definition

We want to show . What are the possible values of for given ? Select all possibilities.

if then

Text. P.111

a a+δ a–δ

Geometrical Interpretation of the Limit Definition, Continued

Use the given graph of f to find a number such that...

if then

Use the given graph of to find a number such that...

if then

Suggested Problems

18)

20)

22)

Textbook p.117

Suggested Problems

18) By the definition of limit, if then .

Hence . Thus as long as , the definition is satisfied, so

20) By the definition of limit, if then

.

Hence . Thus as long as it’s satisfied.

22) By the definition of limit, if then

.

.

Hence . Thus as long as , it’s satisfied.

Textbook p.117

Types of Discontinuities

Figure (a) and (c) are called removable discontinuity because we could remove the discontinuity by redefining f at just the single number 2. Figure (b) is an infinite discontinuity. The discontinuities in figure (d) are called jump discontinuities because the function “jumps” from one value to another.

At which numbers is f discontinuous? Describe the type of discontinuity.

Text p.120

(a) (b) (c) (d)

Suggested Problems

Textbook p.128: Use the definition of continuity to show that the function is continuous at the given number a.

Explain why the function is discontinuous at the given number a.

Find the numbers at which f is continuous. At which of these numbers is f continuous from the right, from the left, or neither?

12) 14)

18) 19)

41)

Suggested Problems - SOLUTIONS -

Textbook p.128: Use the definition of continuity to show that the function is continuous at the given number a.

Explain why the function is discontinuous at the given number a.

Find the numbers at which f is continuous. At which of these numbers is f continuous from the right, from the left, or neither?

12) 14)

18) 19)

41)

(1) (2)

(3)

Thus by the definition of continuity, f(x) is continuous at a = 4.

(1) (2)

(3)

Thus by the definition of continuity, f(x) is continuous at a = 1.

so . Thus by the definition of continuity, f(x) is discontinuous at a = -2

so . Thus by the definition of continuity, f(x) is discontinuous at a = 0

Discontinuous at x = 0. Continuous from the left at x = 0.

Suggested Problems

Text p.129: Find the interval where the function is continuous.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

51) 54)

15) 16)

Suggested Problems - SOLUTIONS -

Text p.129: Find the interval where the function is continuous.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

51) 54)

15) 16)

f(x) is continuous everywhere except at 2

g(x) is continuous on

Let , so and . Then by I.V.Thm, there has to be such that . Thus there is at least one root for the equation.

Let , so and . Then by I.V.Thm, there has to be such that . Thus there is at least one root for the equation.