homework homework assignment #4 read section 2.5 page 91, exercises: 1 – 33 (eoo) quiz next time...

Homework

Homework Assignment #4 Read Section 2.5 Page 91, Exercises: 1 – 33 (EOO) Quiz next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 912. Find the points of discontinuity and state whether f (x) is left- or right-continuous, or neither at these points.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

x

y

Homework, Page 915. (a) For the function shown, determine the one-sided limits at the points of discontinuity.

(b) Which of the discontinuities is removable and how should f be redefined to make it continuous at this point.

The discontinuity at x = 2 is removable by defining f (2) = 6

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

x

y

6

2 4-2

0 0

2

lim , lim 2

lim 6x x

x

f x f x

f x

Homework, Page 91Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous.

By Theorems 2 and 3, respectively, x and sin x are continuous. By Continuity Law ii, 3x and 4 sin x are continuous, and by Continuity Law i, 3x + 4 sin x is continuous

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

9. 3 4sinf x x x

Homework, Page 91Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous.

By Theorem 3, 3x and 4x are continuous. By Theorem 2, 1 + 4x is continuous and by Continuity Law iv, f (x) is continuous

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

313.

1 4

x

xf x

Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.

There is an infinite discontinuity at x = 0.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

117. f x

x

Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.

There is a jump discontinuity at each integer value of x. The function is right-continuous at each jump discontinuity.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

21. f x x

Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.

The function is not defined for x < 0 and it is right-continuous at x = 0.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

3 3225. 3 9f x x x

Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.

The function has a jump discontinuity at x = 2, where it is neither right- nor left-continuous.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2 3 2

29. x x

f xx

Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.

The function is continuous for all values of x.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

33. tan sinf x x

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Chapter 2: LimitsSection 2.5: Evaluating Limits Algebraically

Jon Rogawski

Calculus, ET First Edition

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Examining the graph in Figure 1, it is apparent thatthe value of f(x) approaches8 as x approaches 4. In this section we will look at algebraic methods forevaluating such limits.

2

4

16 16 16 0lim

4 4 4 0x

x

x

0and is undefined as division by zero is undefined.

0

We can not use substitution in this case as substitutionyields:

Indeterminate FormsThe function f (x) has an indeterminate form at x = c if, when f (x) is evaluated at x = c, we obtain an undefined expression of the type:

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

0 00, , 0, , 1 , , 0

0

The function f (x) is also indeterminate at x = c.

If possible, transform f (x) algebraically into a new expression that is defined and continuous at x = c.

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

2

646. lim

9x

x

x

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

3

2

48. lim

2x

x x

x

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

3

2

210. lim

2x

x x

x

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

16

416. lim

16x

x

x

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

20

1 126. lim

x x x x

Example, Page 97 Evaluate the limit or state that it does not exist.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

30. lim sec tan

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

In some cases, such as that shown in Figure 2,the limit at a given point does not exist because the right– and left–hand limits are not equal.

2 2

22 2

lim , lim

lim lim lim D.N. E.x x

xx x

f x f x

f x f x f x

Homework

Homework Assignment #5 Read Section 2.6 Page 97, Exercises: 1 – 49 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company