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Selected Problems on Limits and Continuity
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Selected Problems on Limits and Continuity

Jan 15, 2016

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Selected Problems on Limits and Continuity. 1. lim 2x 3 – 5x 2 + 3x. x 1. 3x 2 – 5x + 2. = lim x(2x 2 – 5x + 3). x  1. 3x 2 – 5x + 2. = lim x (2x – 3)(x – 1). x  1. (3x – 2)(x – 1). = (1) ( 2 (1) – 3). = (1)(2 – 3 ). - 1. (3 – 2 ). ( 3(1) – 2 ). x 2 + 3 + 2. - PowerPoint PPT Presentation
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Page 1: Selected Problems on Limits and Continuity

Selected Problems on Limits and Continuity

Page 2: Selected Problems on Limits and Continuity

1. lim 2x3 – 5x2 + 3xx1 3x2 – 5x + 2

= lim x(2x2 – 5x + 3) x 1 3x2 – 5x + 2

= lim x (2x – 3)(x – 1)(3x – 2)(x – 1)x 1

= (1) ( 2 (1) – 3)

( 3(1) – 2 )= (1)(2 – 3 )

(3 – 2 ) - 1

Page 3: Selected Problems on Limits and Continuity

2. lim x – 1

x2 + 3 - 2 x1● x2 + 3 + 2

x2 + 3 + 2

= lim (x – 1) ( x2 + 3 + 2)x1 x2 + 3 − 4

= lim (x – 1) ( x2 + 3 + 2 )x1 x2 – 1

Page 4: Selected Problems on Limits and Continuity

= lim (x – 1) ( x2 + 3 + 2 )x1 (x + 1)(x – 1 )

= lim ( x2 + 3 + 2 )x1 x + 1

= 12 + 3 + 2 1 + 1

= 4 + 2

2

= 2

Page 5: Selected Problems on Limits and Continuity

3. lim 2x tan 3xx 0

= lim 2x x 0 sin 3x

cos 3x

= lim 2x x 0

● cos 3xsin 3x

1

Page 6: Selected Problems on Limits and Continuity

= lim 2x cos 3x x 0 sin 3x

= lim 2 x 0

● xsin 3x

● cos 3x

= lim 2 x 0

sin 3x● cos 3xx ● 3

3

Page 7: Selected Problems on Limits and Continuity

= lim 2 x 0

sin 3x

3x ● cos 3x● 13

= lim 2 ● 1 ●x 0

13

● cos 3x

= 2 ( 1 ) ( 1 ) cos 3(0)3

= 2 ● cos 03

= 2 ● 1 3

2/3

Page 8: Selected Problems on Limits and Continuity

lim x4 – a4

xa x2 – a2

= lim (x2 + a2) (x2 – a2)x a

x2 – a2

= lim (x2 + a2) x a

= a2 + a2 2a2

4.

Page 9: Selected Problems on Limits and Continuity

5. Find the value of a and b such that

lim a + bx - 3 x 0 x

= 3

Page 10: Selected Problems on Limits and Continuity

lim a + bx - 3 x 0 x

● a + bx + 3

a + bx + 3

lim a + bx - 3 x 0 (x) ( a + bx + 3 )

3

lim 3 + bx - 3x 0 (x) ( 3 + bx + 3 )

= 3

= 3

Page 11: Selected Problems on Limits and Continuity

lim bxx 0 (x) ( 3 + bx + 3 )

lim bx 0 ( 3 + bx + 3 )

b

( 3 + b(0) + 3 )

= 3

= 3

= 3

Page 12: Selected Problems on Limits and Continuity

b

3 + 3

b2 3

= 31

b = 2 9

b = 6 a = 3

= 3

Page 13: Selected Problems on Limits and Continuity

6. The function below is discontinuous at x = 3. Redefine this function to make it continuous.

f (x) = 2x2 – 7x + 3 x – 3

Note: To make a function continuous, we are essentially going to fill up the hole.

Page 14: Selected Problems on Limits and Continuity

f (x) = 2x2 – 7x + 3 x – 3

f(x) = (2x – 1 )(x – 3 )

x – 3 f(x) = 2x – 1

x – 3 = 0

x = 3

f(3) = 2(3) – 1

f(3) = 5

hole: (3, 5)

Page 15: Selected Problems on Limits and Continuity

f (x) = 2x2 – 7x + 3 x – 3

This function is discontinuous at (3,5)

f(x) = 2x2 – 7x + 3 if x 3

5 if x = 3

x – 3

The graph of these two functions are identical except for the hole.

Page 16: Selected Problems on Limits and Continuity

7. The function below is continuous. Find the value of the constant c.

f(x) = x2 – c2 if x < 4

cx + 20 if x 4

Since each part of these function is a polynomial function, then each of them is continuous. Thus the only possible point of discontinuity is at the boundary point.

Page 17: Selected Problems on Limits and Continuity

lim x2 – c2

42 – c2

lim cx + 20

c(4) + 2042 – c2 = 4c + 2016 – c2 = 4c + 20

16 – c2 – 4c – 20 = 0- c2 – 4c – 4 = 0c2 + 4c + 4 = 0 (c + 2)(c + 2) = 0

c = - 2

x4 x4

Page 18: Selected Problems on Limits and Continuity

8. Find the values of a and b such if the function f below is continuous.

f(x) = ax2 + x – b if x 2

2x + b if 2 < x < 5

2ax – 7 if x 5

Page 19: Selected Problems on Limits and Continuity

a(2)2 + 2 – b 2(2) + b

4a + 2 – b 4 + b

4a + 2 – b = 4 + b

4a + 2 – b – 4 – b = 0

4a – 2b - 2 = 0

lim ax2 + x – b lim 2x + b x2 x2

Page 20: Selected Problems on Limits and Continuity

8. Find the values of a and b such if the function f below is continuous.

f(x) = ax2 + x – b if x 2

2x + b if 2 < x < 5

2ax – 7 if x 5

Page 21: Selected Problems on Limits and Continuity

lim 2ax - 7lim 2x + b

2(5) + b 2a(5) – 7

10 + b = 10a – 7

b = 10a – 7 – 10

b = 10a - 17

2a – 2b - 2 = 0

4a – 2 (10a – 17 ) - 2 = 0

x5 x5

Page 22: Selected Problems on Limits and Continuity

4a – 20a + 34 - 2 = 0

- 16a + 32 = 0

- 16a = - 32a = - 32

- 16a = 2

b = 10a - 17

b = 10(2) – 17

b = 20 – 17

b = 3

Page 23: Selected Problems on Limits and Continuity

9. Find the value of a such that the following is continuous.

f(x) = ax if x 0tan x

x2 – 2 if x < 0

Page 24: Selected Problems on Limits and Continuity

lim axx0 tan x

=lim x2 - 2

x0

? = - 2

Page 25: Selected Problems on Limits and Continuity

lim ax x 0 tan x

axsin xcos x

ax ● cos x sin x

ax cos x sin x

lim x0

lim x0

=

lim x0

lim x0

a ● x ● cos x sin x

Page 26: Selected Problems on Limits and Continuity

lim x0

a ● 1 ● cos x

= a ( 1 ) ( cos 0)

= a ( 1 ) ( 1 )

= a

Page 27: Selected Problems on Limits and Continuity

lim axx0 tan x

= lim a2 - 2x0

? = - 2

a = - 2

Page 28: Selected Problems on Limits and Continuity

10. Given that lim f(x) = 5 x3

lim g(x) = 0x3

lim h(x) = - 8x3

find the following;

a) lim ( f(x) + g(x) )x3

= lim f(x) + lim g(x) x3 x3

= 5 + 0 = 5

Page 29: Selected Problems on Limits and Continuity

10. Given that lim f(x) = 5 x3

lim g(x) = 0x3

lim h(x) = - 8x3

find the following;

b) lim ( f(x) / h(x) )x3

= lim f(x) / lim h(x) x3 x3

= 5/-8

Page 30: Selected Problems on Limits and Continuity

10. Given that lim f(x) = 5 x3

lim g(x) = 0x3

lim h(x) = - 8x3

find the following;c) lim 2 h(x)

x3 f(x) – h(x)

= 2 lim h(x)x3

lim f(x) - lim h(x)x3 x3

Page 31: Selected Problems on Limits and Continuity

= 2 ( - 8 )

5 – ( - 8 )

= - 1613

Page 32: Selected Problems on Limits and Continuity

10. Given that lim f(x) = 5 x3

lim g(x) = 0x3

lim h(x) = - 8x3

find the following;

d) lim x2 f(x)x3

= lim x2 x3 x3

= 32 ● 5 = 45

● lim f (x )