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Section 2.8 - Continuity 2.2.

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A function f(x) is continuous at x = c if and only if all three of the following tests hold: f(x) is right continuous at x = -5 f(x) is continuous at x = -4 f(x) has infinite discontinuity at x = -3 [i, iii] f(x) has point discontinuity at x = -2 [i, iii] f(x) has infinite discontinuity at x = -1 [i, ii, iii] f(x) is continuous at x = 0
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Section 2.8 - Continuity 2.2
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Page 2: Section 2.8 - Continuity 2.2.

A function f(x) is continuous at x = c if and only if all three of the following tests hold:

i. f c exists

x c

ii. lim f x exists

x c

iii. lim f x f c

f(x) is right continuous at x = -5

f(x) is continuous at x = -4

f(x) has infinite discontinuity at x = -3 [i, iii]

f(x) has point discontinuity at x = -2 [i, iii]

f(x) has infinite discontinuity at x = -1 [i, ii, iii]

f(x) is continuous at x = 0

Page 3: Section 2.8 - Continuity 2.2.

At x = 1

At x = 2

At x = 3

At x = 4

At x = 5

Point Discontinuity [i, iii]

Jump Discontinuity [i, ii, iii]

Continuous

Continuous

Point Discontinuity [i, (ii), iii]

Page 4: Section 2.8 - Continuity 2.2.

2f x x 2x 1

continuous

2

1f xx 1

continuous

2

xf xx x

xx x 1

pt. discontinuity at x = 0inf. discontinuity at x = 1

2

x 3f xx 9

x 3

x 3 x 3

pt. discontinuity at x = 3inf. discontinuity at x = -3

2

2x 3 x 1f x

x x 1

2 1 3 1

21 1

continuous

1 x 1 x 2

f x 23 x x 2

1 2 1 22

3 2 1

jump discontinuity at x = 2

Page 5: Section 2.8 - Continuity 2.2.

Find the value of a which makes the function below continuous

3

2

x x 2f x

ax x 2

3

2

3

2

x x 2f x

ax x

2 8

2 4a2 a

4a 8 a 2

Page 6: Section 2.8 - Continuity 2.2.

Find (a, b) which makes the function below continuous

2 x 1

f x ax b 1 x 32 x 3

As we approach x = -1 2 = -a + b

As we approach x = 3 -2 = 3a + b

4 4aa 1 b 1

1,1

Page 7: Section 2.8 - Continuity 2.2.

Consider the function sinx x 0

f x xk x 0

Find the value of k which makes f(x) continuous at x = 0

Sincex 0

sinxlim 1x

, if k =1, the hole is filled.

Page 8: Section 2.8 - Continuity 2.2.

Calculator RequiredLet m and b be real numbers and let the function f be defined by:

21 3bx 2x x 1

f xmx b x 1

If f is both continuous and differentiable at x = 1, then:A. m 1, b 1B. m 1, b 1C. m 1, b 1D. m 1, b 1E. none of these

3b 4x x 1f ' x

m x 1

3b 4 m

1 3b 2 m b

3b m 42b m 3

Page 9: Section 2.8 - Continuity 2.2.

No CalculatorThe function f is continuous at x = 1

x 3 3x 1 x 1If f x then kx 1

k x 1

1 1A. 0 B.1 C. D. E. none of these2 2

xx 3 3x 1 3 3x 1xx 31 3

f xx 1

x 3 3x 1

3 3x 1x x1

x 1

2 x 1

x 3 3x 1

Page 10: Section 2.8 - Continuity 2.2.

Calculator Required

Which of the following is true about 2

2

x 1f x ?

2x 5x 3

I. f is continuous at x = 1 II. The graph of f has a vertical asymptote at x = 1III. The graph of f has a horizontal asymptote at y = 1/2

A. I B. II C. III D. II, III E. I, II, III

I. f(1) results in zero in denominator….NO

II. Since x – 1 results in 0/0, it is a HOLE, NOT asymptote

III. 2

2x

x 1lim

2x 5x 3

2

2x

x 2x 1lim2x 5x 3

X XX X

12

Page 11: Section 2.8 - Continuity 2.2.

No CalculatorWhich function is NOT continuous everywhere?

2 / 3

2

2

2

A. y x

B. y x

C. y x 1xD. y

x 14xE. y

x 1

undefined at x = -1

Page 12: Section 2.8 - Continuity 2.2.

The graph of the derivative of a function f is shown below.Which of the following is true about the function f? I. f is increasing on the interval (-2, 1) II. f is continuous at x = 0III. f has an inflection point at x = 2

A. I B. II C. III D. II, III E. I, II, III

NOYESNO

Calculator Required

Page 13: Section 2.8 - Continuity 2.2.

tanxConsider the function f defined on x by f x2 2 sinx

No Calculator

for all x . If f is continuous at x = , then f

A. 2 B. 1 C. 0 D. -1 E. -2

sinx

tanx cos xf xsinx sinx

sinx 1cos x sinx

1 1 1 Dcosx cos


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