Limit & Derivative Problems

Problem… Answer and Work…

1.

1. 2

2 4

3 7 11lim

2 4 8x

x x

x x x

1

2

4 4 4

2 4

4 4 4

3 7 110

lim 082 4 8x

x x

x x xx x x

x x x

Limit & Derivative Problems

Problem… Answer and Work…

2.

2.

16

16lim

4x

x

x

2

16 16

4 4lim lim 4 16 4 8

4x x

x xx

x

Limit & Derivative Problems

Problem… Answer and Work…

3.

3. 3

2

8lim

2x

xx

3

2

2

2 2

2

2 2 4lim

2lim 2 4 2 2 2 4

4 4 4

12

x

x

x x x

xx x

Limit & Derivative Problems

Problem… Answer and Work…

4. Consider the function given by

Is f(x) continuous at x=1? Justify.

4.

2 3, x 1( )

3x, x > 1

xf x

4

12

1

1

Does lim ( ) (1)?

lim ( ) 1 3 4

lim ( ) 3 1 3

lim not equal to each other,

theref ore,

not continuous at x = 1

x

x

x

f x f

f x

f x

Limit & Derivative Problems

Problem… Answer and Work…

5. Is the function given by

continuous for all x? If not, where are the discontinuities? Are they removable?

5.

2, x 3( )

6 - x, x 3

xh x

5

3

3

3

3 3

lim ( ) ( )?

lim ( ) 3 2 5

lim ( ) 6 3 3

not continuous at x = 3

lim ( ) lim ( )

shows that the discontinuity

is not removable

x

x

x

x x

h x h x

h x

h x

h x h x

Limit & Derivative Problems

Problem… Answer and Work…6. Let the piecewise function f be

defined as follows:

Which of the following is true about the function f?

I. f(2) = 2

II.

III. f(x) is continuous at x = 2

A. I only

B. III only

C. I and II only

D. I and III only

E. I, II, and III

6. Test: f(2) = 2? Yes, so I is true

Test:

Test: f(x) is continuous at x = 2?

Does the lim f(x) = f(2)?

4 is not equal to 2

No, so III is false

Answer is A) I only

2 4, f or x 2

( ) 22, f or x = 2

xf x x

6

2lim ( ) 2x

f x

2

22

22

lim ( ) 2?

( 2)( 2)lim ( ) lim

( 2)lim ( ) lim 2 4

No, I I is f alse

x

xx

xx

f x

x xf x

xf x x

Limit & Derivative Problems

Problem… Answer and Work…

7.

What is the value of a for which f(x) is continuous for all values of x?

A. -2

B. -1

C. 0

D. ½

E. 1

7. To be continuous at x = 1

7

2

1, x 1I f ( )

3 , x > 1

xf x

ax

1 12

1 12

2

lim ( ) lim ( )

lim 1 lim3

2 = 3 + ax

2 = 3 + a(1)

2 = 3 + a

-1 = a

x x

x x

f x f x

x ax

Limit & Derivative Problems

Problem… Answer and Work…

8. Find the cartesian coordinates of the point on the graph of

where the instantaneous rate of change of f is equal to 5

8.

to find y substitute x = ½ in the original function f(x)

Ans: (1/2, 11/4)

8

2( ) 3 2 1f x x x

2( ) 3 2 1f x x x

' ( ) 6 2

5 6 2

3 6

12

f x x

x

x

x

Limit & Derivative Problems

Problem… Answer and Work…

9. Which of the following directly describes the discontinuities associated with

a. A hole at x = 3, a vertical asymptote at x = 3

b. Holes at x = -3 and x = 3

c. A hole at x = 3, a vertical asymptote at x = -3

d. Vertical asymptotes at x = 3 and x = -3

e. No discontinuities

9.

Hole at x = 3 because we factored out (x – 3)

There is a vertical asymptote at x = -3

9

2

2

2 3( )

9

x xf x

x

( 3)( 1) 1( 3)( 3) 3x x xx x x

Limit & Derivative Problems

Problem… Answer and Work…10. Given the piecewise function

For what values of a and b is f(x) differentiable at x = 1?

A. a = 2 b = -3

B. a = 2 b = -2

C. a = -2 b = 1

D. a = 3 b = -1

E. a = 5 b = 8

10. Differentiability implies continuity

To be differentiable x = 1

Solve for a when b = 1

a – 1 = -3 a = -2 Ans: C

10

2

2 x 1( )

bx 1 x > 1

x af x

2

2

2 1 when x = 1

2(1) + a = b(1) 1

2 1

3

x a bx

a b

a b

2(2 ) ( 1)

2 2 when x = 1

2 2 (1)

1

d dx a bx

dx dxbx

b

b

Limit & Derivative Problems

Problem… Answer and Work…

11. Which of the following is (are) true about the function

I. It is continuous at x = 0II. It is differentiable at x = 0III.

A. I onlyB. II onlyC. I and III onlyD. II and III onlyE. I, II, III

11. Test 1: Continuous at x = 0

yes

Test 2: Differentiable at x = 0?

No

Test 3:

Yes

Ans: C

11

13( ) ?f x x

0lim ( ) 0x

f x

3( )f x x

2 3

2 3

1'( )

31

03

undefi ned at x = 0

f x x

x

0

0 0

lim ( ) 0?

lim ( ) 0 lim ( ) 0x

x x

f x

f x f x

Limit & Derivative Problems

Problem… Answer and Work…12. To apply either the Mean Value

Theorem or Rolle’s Theorem to a function f, certain requirements regarding the continuity and differentiability of the function must be met. Which of the following states the requirements correctly?

A. f is continuous on (a, b) and differentiable on (a, b)

B. f is continuous on (a, b) and differentiable on [a, b]

C. f is continuous on (a, b) and differentiable on [a, b)

D. f is continuous on [a, b] and differentiable on (a, b)

E. f is continuous on [a, b] and differentiable on [a, b]

12. Look at the definition of Rolle’s Theorem and the Mean Value Theorem

f is continuous on [a, b] and differentiable on (a, b)

Ans: D

12

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

A. Determine the x and y intercepts, if any. Justify your answer.

13. A

13

2

1 1( ) 1f x

x x

2

2

2

2

2

2

2

1( )

10

0 1 no real solutions

n

x-intercept

y-intercep

o x-intercepts

0

t

0 1 undefi ned

0no y-intercepts

x xf x

x

x x

xx x

y

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

B. Write an equation for each vertical and each horizontal asymptote. Justify your answer.

13. B

Vertical asymptote

Horizontal asymptote

14

2

1 1( ) 1f x

x x

2 0

0

x

x

2

2 2 2 2

2 20 0

2

11

lim lim 1

1

x x

x xx x x x x

x x

xy

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

C. Determine the intervals on which f is increasing or decreasing. Justify your answer.

13. C

15

2

1 1( ) 1f x

x x

1 2

2 3

3

3

( ) 1

2'( ) 2

0 ( 2)

0 x = -2

Do a sign graph f or the critical points 0, -2

0 is a vertical asympt

Decreasi

ote

ng (- , -2) and (0, )

I ncreasing (-2, 0)

f x x x

xf x x x

x

x x

x

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

D. Determine the relative minimum and maximum points, if any. Justify your answer.

13. D

Relative minimum occurs at x = -2

when x = -2

16

2

1 1( ) 1f x

x x

1 2( 2) 1 ( 2) ( 2)

1 1 31

2 4 43

( 2, )4

no maximum because at x = 0 which

is the vertical asymptote

f

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

E. Determine the intervals on which f is concave up or concave down. Justify your answer.

13. E

17

2

1 1( ) 1f x

x x 2 3

3 4

4

' ( ) 2

"( ) 2 6

2 ( 3)

undefi ned at x=0 because

it is a vertical asymptote

0 = x+3

3

Do a sign graph using critical

Concaves d

points -3,

own (- , -3)

Concaves up (-3,0) (0, )

0

f x x x

f x x x

x x

x

Limit & Derivative Problems

Problem… Answer and Work…13. Let f be the function defined by

F. Determine any points of inflection

13. F

Point of inflection when x = 3

18

2

1 1( ) 1f x

x x

1 2

2

( 3) 1 ( 3) ( 3)

1 11

7( 3, )

9

3 ( 3)9 3 19 9 979

f

Limit & Derivative Problems

Problem… Answer and Work…14. On the interval [1, 3], what is the

average rate of change for the functions, if

14.

19

2( ) 3 4 ?s t t t

2

2

(3) (1)3 1

(3) 3(3) 4(3) 27 12 15

(1) 3(1) 4(1) 3 4 1

15 ( 1)8

3 1

s sAvg

s

s

Limit & Derivative Problems

Problem… Answer and Work…15. Is the function defined by

continuous at x = 4? Justify your answer.

15.

20

3, 3 x < 7( )

5, 7

xf x

x x

4

4

lim 3 4 3 1

(4) 4 3 1

lim ( ) (4)

x

x

x

f

f x f