Homework

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

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Example, Page 97 Show that the limit leads to an indeterminate form. Then transform the function algebraically and evaluate.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

5

251. lim

5x

x

x

22

5

2

5 5 5

2

5

5 2525 0lim

5 5 5 05 525

lim lim lim 5 105 5

25lim 10

5

x

x x x

x

x

xx xx

xx 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

8

645. lim

8x

x

x

2

8 8 8

2

8

8 864lim lim lim 8 16

8 8

64lim 16

8

x x x

x

x xxx

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

32

29. lim

4x

x

x x

3 22 2 2

2

32

2 2 2lim lim lim

2 24 4

1 1 1lim

2 2 2 2 8

2 1lim

84

x x x

x

x

x x x

x x xx x x x

x 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

22

3 4 413. lim

2 8x

x x

x

2

22 2

2

2

22

3 2 23 4 4lim lim

2 2 22 8

3 2 23 2 8lim 1

2 2 2 2 2 8

3 4 4lim 1

2 8

x x

x

x

x xx x

x xx

x

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

0

1 13 317. lim

h

hh

0 0 0

0

1 13 3 3 33 3lim lim lim3 3 3 3 3 3

1 1lim

3 3 9

h h h

h

h h hhh h h h h h

h

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

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

221. lim

4x

x

x x

2 2

2

2

2 42 4lim lim

44 4

2 4lim

2 2

4lim 2

2

x x

x

x

x x xx x x

x xx x x x

x x 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

4

1 425. lim

42x xx

4 4

4 4

4

1 4 1 2 4lim lim

4 42 2 2

2 4 2lim lim

4 4

2 2lim

4 2

x x

x x

x

x

x xx x x

x x

x x

x x

x x

Example, Page 97 25. Continued.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

4 4

4

4

1 4 2 2lim lim

4 42 2

4lim

4 2

1 1lim

42

x x

x

x

x x

x xx x

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

4

sin cos29. lim

tan 1x

x x

x

4 4

4

4

sin cos cossin cos coslim lim

tan 1 cos sin cos

2lim cos

2

sin cos 2lim

tan 1 2

x x

x

x

x x xx x x

x x x x

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

3

2cos 3cos 233. lim

2cos 1x

x x

x

2

3 3

3

2

2

2cos 1 cos 22cos 3cos 2lim lim

2cos 1 2cos 1

1 5lim cos 2 2

2 2

2cos 3cos 2 5lim

2cos 1 2

x x

x

x

x xx x

x x

x

x x

x

Example, Page 97 Use the identity: a3 – b3 =(a – b)(a2 + ab + b2).

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

3

1

137. lim

1x

x

x

23

2

1 1 1

3

1

1 11lim lim lim 1

1 13

1lim 3

1

x x x

x

x x xxx x

x x

x

x

Example, Page 97 Use the identity: a3 – b3 =(a – b)(a2 + ab + b2).

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

4

31

141. lim

1x

x

x

24

3 21 1

2

21

4

31

1 1 11lim lim

1 1 1

1 1 4lim

31

1 4lim

31

x x

x

x

x x xx

x x x x

x x

x x

x

x

Example, Page 97 Evaluate the limit in terms of the constants involved.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

1

45. lim 4 2 3t

t at a

1

1

lim 4 2 3 4 1 2 1 3

4 2 3 5 4

lim 4 2 3 5 4

t

t

t at a a a

a a a

t at a a

Example, Page 97 Evaluate the limit in terms of the constants involved.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

49. limx a

x a

x a

lim lim

1 1lim

2

1lim

2

x a x a

x a

x a

x a x a x a

x a x a x a x a

x a a

x a

x a a

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Chapter 2: LimitsSection 2.6: Trigonometric Limits

Jon Rogawski

Calculus, ET First Edition

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Figure 1 shows a “trapped”function f (x) such that l (x) ≤ f (x) ≤ u (x) for all x

Figure 2 shows a “squeezed”function f (x) such that l (x) ≤ f (x) ≤ u (x) for x ≠ cand l (c) = f (c) = u (c).

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Figure 3 illustrates how the function y = x sin (1/x) is squeezed as x approaches 0.Mathematically,

0 0 0

1sin

1 1sin 1 sin

,

lim lim 0 lim 0x x x

f x xx

x xx x

u x x l x x

l x f x u x

l x u x f x

Example, Page 102Use the Squeeze Theorem to evaluate the limit.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

0

16. lim sin

xx

x

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

The above diagram leads us to the following conclusion:1 1 1

sin tan sin tan2 2 2

sin sinsin sin 1

cos

sin sin sincos cos 1

cos

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

From the results on the previous slide, we state Theorem 3.

The graph of Theorem 3 isshown in Figure 4.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

The first part of Theorem 2 follows from the “squeeze” citedin Theorem 3. Part 2 results from:

2 2

0 0 0

0 0 0

0

1 cos 1 1 cos 1 sinlim lim lim

1 cos 1 cos1 sin 1

lim limsin lim 0 1 01 cos 1 1

1 coslim 0

x x x

x x x

x

Example, Page 102Evaluate the limit using Theorem 2 as necessary.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

0

tan12. lim

x

x

x

Example, Page 102Evaluate the limit.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

026. lim

csc25x

x

x

Example, Page 102Evaluate the limit.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

0

sin3 sin 234. lim

sin5x

x x

x x

Homework

Homework Assignment #6 Read Section 2.7 Page 102, Exercises: 1 – 45 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company