Limits and Continuity - Thornton Fractionalmoodle.tfd215.org/pluginfile.php/64947/mod_resource/content/1/01... · Limits and Continuity . Big Ideas . ... 2008 BC Multiple Choice

Limits and Continuity Big Ideas Need limits to investigate instantaneous rate of change Do not care what the function is actually doing at the point in question Limits may exist at a point even if the function itself does not exist at that point No reason to think that the limit will have the same value as the function at that point Not all limits will exist Limit is the height that the function intends to reach Limits at a point Limits at infinity Infinite limits Infinity is not a number, but rather indicates a direction Vertical and horizontal asymptotes

Vertical asymptotes: lim ( ) then is a vertical asymptote

Horizontal asymptotes: lim ( ) then is a horizontal asymptotex a

x

f x x a

f x a y a→

→±∞

= ±∞ =

= =

One-sided limits lim ( ) iff lim ( ) and lim ( )

x c x c x cf x L f x L f x L

− +→ → →= = =

Continuity lim ( ) ( ) Must show three conditions are met

x cf x f c

→=

Discontinuity – Removable (limit exists) Non-removable (limit does not exist: jump, infinite, oscillating) Sandwich Theorem (Squeeze Theorem)

Indeterminate forms: 0 00 0 1 00

∞∞∞ ∞ −∞ ∞

∞

Determinate Forms: 00 00

k kk k∞

→ →∞ →∞ → ∞+∞→∞ ∞ ∞→∞∞

Properties of limits End Behavior

Mike Koehler 1 - 1 Limits and Continuity


AP Multiple Choice Questions 2008 AB Multiple Choice Problems 1 5 77 2008 BC Multiple Choice Problems 3 78 2003 AB Multiple Choice 3. For 0x ≥ , the horizontal line 2y = is an asymptote for the graph of the function f . Which of the

following statements must be true? A) (0) 2f = B) ( ) 2 for all 0f x x≠ ≥ C) (2) is undefinedf

D) 2lim ( )x

f x→

= ∞ E) lim ( ) 2x

f x→∞

= 6. 3 2

3 2

2 3 4lim4 3 2 1x

x x xx x x→∞

− + −=

− + −

A) 4 B) 1 C) 14

D) 0 E) -1

79. For which of the following does

4lim ( ) existx

f x→

?

A) I only B) II only C) III only D) I and II only E) I and III only


2003 BC Multiple Choice 81. The graph of the function f is shown in the figure on

the right. The value of ( )1

limsin ( )x

f x→

is

A) 0.909 B) 0.841 C) 0.141 D) -0.416 E) nonexistent

1998 AB Multiple Choice 26. The function f is continuous on the closed interval

[ ]0,2 and has values that are given in the table on the

right. The equation 1( )2

f x = must have at least two

solutions in the interval [ ]0,2 if k =

x 0 1 2 ( )f x 1 k 2

A) 0 B) 12

C) 1 D) 2 E) 3

76. The graph of a function f is shown in the figure on

the right. Which of the following statements aboutf is false?

A) is continuous at f x a= B) has a relative maximum at f x a=

C) is in the domain of x a f= D) lim ( ) is equal to lim ( )x a x a

f x f x+ −→ →

E) lim ( ) existsx a

f x→


83. 2 2

4 4If 0, then lim is x a

x aax a→

−≠

−

A) 2

1a

B) 2

12a

C) 2

16a

D) 0 E) nonexistent

1997 AB Multiple Choice 15. The graph of the function f is shown in the figure

on the right. Which of the following statements about f is true?

A) lim ( ) lim ( )

x a x bf x f x

→ →= B) lim ( ) 2

x af x

→= C) lim ( ) 2

x bf x

→=

D) lim ( ) 1x b

f x→

= E) lim ( ) does not existx a

f x→

21.

1lim is

lnx

xx→

A) 0 B) 1e

C) 1 D) e E) nonexistent


AP Free Response Questions 2012 AB 4 Part c (continuity) The function f is defined by 2( ) 25 for 5 5f x x x= − − ≤ ≤ . c)

Let g be the function defined by( ) for 5 3

( )7 for 3 5

f x xg x

x x− ≤ ≤ −

= + − < ≤.

Is g continuous at 3x = − ? Use the definition of continuity to explain your answer. 2011 AB 6 Part a (continuity)

Let f be a function defined by 4

1 2sin for 0( ) .

for 0x

x xf x

e x−

− ≤=

>

a) Show that f is continuous at 0.x =

1998 AB 2 Part a Let f be the function given by 2( ) 2 xf x xe= . a) Find lim ( ) and lim ( )

x xf x f x

→−∞ →∞.

1978 AB 3 Parts a b

Given the function 2

2 2 defined by ( )2

xf f xx x

−=

+ −

a) For what values of is ( )x f x discontinuous? b) At each point of discontinuity found in part (a), determine whether ( )f x has a limit, and if so give the value

of the limit. 1971 AB 6 Part c A particle starts at the point ( )5,0 at 0t = and moves along the x-axis in such a way that at time 0t > its velocity

( )v t is given by 2( )1

tv tt

=+

.

c) Find the limiting value of the velocity as t increased without bound.


Textbook Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l2012

Section Questions 2.1 43 44 45-50 55 56 57-60 2.2 53 54 55 56 QQ p 77 1 2 3 2.3 11-18 19-24 47-50 2.4 29 30 36 QQ p 95 1 2 3 4

Handouts



AP Calculus Chapter 2 – Section 1 Limits 1. The graphs of and f g are given below.

Graph of f

Graph of g

Determine whether the following limits exist. If they do, then find the limit. a. ( )

1limx

f x→−

b. ( )1

limx

f x→

c. ( )

1limx

g x→−

d. ( )1

limx

g x→

e. ( ) ( )

1lim x

f x g x→−

+ f. ( ) ( )0

lim 2 3x

f x g x→

+

g. ( ) ( )

1lim x

f x g x→−

h. ( ) ( )2

lim x

f x g x→

i. ( )( )0

lim x

f xg x→

j. ( )( )0

lim x

g xf x→

k. ( )( )

2lim x

g f x→−

l. ( )( )1

lim x

f g x→−

2. The graphs of functions f and g are those given in problem 1 above. Determine whether the following limits exist and find the limit when it exists. a. ( )

1lim

xf x

−→− b. ( )

1lim

xf x

+→−

c. ( )

1lim

xg x

−→− d. ( )

1lim

xg x

+→−

e. ( )

0lim 2x

f x−→

+ f. ( )2

1lim

xf x

−→−


Answers Problem 1 a DNE

b DNE

c 1

d -1

e DNE

f 4

g DNE

h 0 Hint: take left and right hand limits

i DNE

j 0

k 0

l 2 Hint: take left and right hand limits

Problem 2 a 1

b 2

c 1

d 1

e 1

f 0


AP Calculus Chapter 2 Section 1 Limits

Refer to the graph above to answer each of the following questions. If a limit does not exist explain why.

0

00

1. lim ( ) 2. lim ( ) 3. lim ( )

4. lim ( ) 5. lim ( ) 6. lim ( )

7. lim ( ) 8. lim ( ) 9. lim ( )

10. lim ( ) 11. lim ( ) 12. lim ( )

13. lim ( ) 14. lim ( ) 15. l

x x x a

x ax a x b

x bx b x

xx x c

x cx c

f x f x f x

f x f x f x

f x f x f x

f x f x f x

f x f x

−

+ −

+ −

+ −

+

→−∞ →∞ →

→→ →

→→ →

→→ →

→→

= = =

= = =

= = =

= = =

= = im ( )

16. lim ( ) 17. lim ( ) 18. lim ( )

19. ( ) 20. ( ) 21. ( )

x d

x d x ex d

f x

f x f x f x

f b f d f e

−

+

→

→ →→

=

= = =

= = =

22. How does the answer to number 19 compare to the answer to number 8?

Graph of f(x)


Answers 1 n 2 k 3 ∞ 4 m 5 Does not exist 6 0 7 −∞ 8 Does not exist 9 h 10 h 11 h 12 p 13 m 14 Does not exist 15 0 16 m 17 Does not exist 18 0 19 m 20 0 21 0 22 While ( )f b m= the limit as x approaches b does not exist because the left and right hand limits are not equal.


AP Calculus Chapter 2 Section 1 Trigonometric Limits Find the following limits involving trigonometric functions.

( )2

0

0

0

0

2

0

0

2

20

0

1.

3.

5.

7.

sin 2

1 sinlim1 cos

2. limsin3

tan 2lim

sin 24. lim4

sinlim3

sin6. lim1 cos

lim1 cos

8. lim

x

x

x

x

x

x

x

x

x

x

xx

xx

xx

xx

xx

x xx

xx

→

→

→

→

→

→

→

→

++

⋅−

−


Answers: 1 11. 2.2 3

13. 2 4.2

5. 0 6. 27. 1 8. 0

How to work:

0

0 0

0 0

0 0

2

0 0

1 0 11.1 1 2

1 1 113 3 3

1 2 13. 1 2 2cos(2 ) 1 1

1 1 112 2 2

sin 15. 1 01 3

1 sinlim1 cos

32. lim limsin3 sin3

tan 2 sin 2lim lim2

sin 2 sin 24. lim lim4 2

sin sinlim lim3

x

x x

x x

x x

x x

x

x

xx

x xx x

x xx x

x xx x

x xx x

→

→ →

→ →

→ →

→ →

+= =

+

= ⋅ = ⋅ =

= ⋅ ⋅ = ⋅ ⋅ =

= ⋅ = ⋅ =

= ⋅ ⋅ = ⋅ ⋅

++

⋅

( )

2

2

0 0 0 0 0

2 2

2 20 0 0

0

13

1 cos sin (1 cos ) (1 cos ) (1 cos ) 1 2 21 cos sin sinsin

7. 1 1 1

sin 2

0

sin sin6. lim lim lim lim lim1 cos 1 cos

lim lim limsin sin1 cos sin

8. lim

x x x x x

x x x

x

x x x x x x x xx x xx

x

x

x x x xx x

x x x xx xx x

→ → → → →

→ → →

→

+ ⋅ ⋅ + +⋅ = = = + = ⋅ =+

= = ⋅ = ⋅ =

=

=

⋅ ⋅=− −

−

( ) ( )2 2

20 0

sin 2 sin 22 2 1 0 02 12

lim limx x

x xx xx x x→ →

⋅ = ⋅ = ⋅ =


AP Calculus Chapter 2 Sections 1 – 3 Limits 1. Use the table feature of the calculator to fill in the table and guess the value of the limit of the function ( )f x .

3

21

1lim1x

xx→

−−

x ( )f x x ( )f x 1.002 0.998 1.001 0.999 1.0005 0.9995 1.00001 0.99999

2. Evaluate the limit assuming that

4 4lim ( ) 3 and lim ( ) 2x x

f x g x→ →

= = .

4 4

24 4

2

4 4

lim ( ) ( ) lim(2 ( ) 5 ( ))

( ) ( ) 1lim lim3 ( ) 9

( )lim ( ) lim4

x x

x x

x x

f x g x f x g x

g x f xg xx

g xf xx

→ →

→ →

→ →

+

+−

−

3. Evaluate the limits algebraically or state that the limit does not exist.

( )

32 2

0 0

2 2 2

0 8

3 2 2lim lim2 4

1 12 2 3 3lim lim

3 9 64lim lim9

x x

x x

x x

x x xx x x

x xx x

a x a xx x

→ →

→ →

→ →

− + −− −

−+ − +

+ − −−


4. Limits involving trigonometric functions. Find the following limits or state that they do not exist.

0 0

0 0

0 0

sin cos sin(6 )lim lim

sin(7 ) sin(2 )lim lim3 sin(5 )

1 sin(3 )sin(2 )lim limsin sin(5 )

x x

x x

x x

x x xx x

x xx x

x xx x x

→ →

→ →

→ →

5. Limits involving infinity. Find the following limits. Indicate if the function goes to ±∞ .

4 2

2

2lim lim 2int( ) 34

3 sin( )lim lim4 2

x x

x x

x xx

x x xx x

− −→ →

→∞ →∞

−−

+ +−

6. Use limits to find the vertical asymptote(s) of the function 2

2

6( )2

x xf xx x+ −

=− −

.

7. Use limits to find the horizontal asymptote(s) of the function 3 5( )2 4

xf xx−

=+

.

8. Determine the points at which the function is discontinuous and state the type of discontinuity: removable,

jump, infinite, or none of these.

2

1 3( ) ( )2 9

2 2( ) ( )1 2

xf x f xx x

x xf x f xx x

−= =

− −

− −= =

− −


9. Is the function ( )f x continuous? Give reasons for your answer.

1 1

( ) 1 1

x xf x

xx

+ <=

≥

10. Find the value of c that makes the function continuous.

2 5

( )4 2 5x c x

f xx c x

− <=

+ ≥

11. Intermediate Value Theorem

Use the IVT to show that 3( )f x x x= + takes on the value 9 for some x in [ ]1,2 . The temperature in Kansas City was 46o at 6:00 AM and rose to 68o at noon. What must we assume about

temperature to conclude that the temperature was 60o at some moment between 6:00AM and noon. 12. Draw the graphs of a function on [ ]0,5 with the given properties.

1 3 3

1 3 3

42 2

lim ( ) 2, lim ( ) 0, lim ( ) 4

lim ( ) , lim ( ) 0, lim ( )

lim ( ) (2) 3, lim ( ) 1, lim ( ) 2 (4)

( ) has a removable discontinuity at 1, a jump discontinuity at

x x x

x x x

xx x

f x f x f x

f x f x f x

f x f f x f x f

f x x x

− +

− +

+ −

→ → →

→ → →

→→ →

= = =

= ∞ = = −∞

= = = − = ≠

=3 3

2,and lim ( ) , lim ( ) 2x x

f x f x− +→ →

= = −∞ =



AP Calculus Chapter 2 Section 4 1. A curve has equation ( )y f x= .

Write an expression for the slope of the secant line through the points (3, (3))P f and (3 , (3 ))Q h f h+ + . Write an expression for the slope of the tangent line at P. 2. Consider the slope of the given curve at each of the

five points shown. List these five slopes in decreasing order and explain

your reasoning.

3. Shown at the right are the position functions of two

runners, A and B, who run a 100-meter race and finish in a tie.

Runner A is the top graph and runner B is the lower graph.

Describe and compare how the runners run the race. At what time is the distance between the runners the

greatest? At what time do they have the same speed?


In Problems 4 to 7, use the formula 0

( ) ( )limh

f a h f amh→

+ −= to find the slope of the tangent line.

4. Find the slope of the tangent to the parabola 2 2y x x= + at the point ( )3,3− . Use the results to write the equation of the tangent line.

5. Find the slope of the tangent to the curve 23 4 2y x x= + − at the point (2,3) .

Find the equation of the tangent line. Find the equation of the normal line.

6. Find the slope of the tangent to the curve 12

yx

= at the point 12,4

.

Find the equation of the tangent line. 7. Find the equation of the tangent to the curve 2( ) 2 3f x x= − when the slope of the tangent is 12. 8. Find the average rate of change of the function ( ) 2f x x= + over the interval [ ]2,23 . 9. An object dropped from the top of a 200-foot cliff will fall 216s t= feet in t seconds.

How far will the object fall in 3 seconds?

What is the average speed over these 3 seconds?

What is the instantaneous speed at 3t = seconds?


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Limits and Continuity - Thornton Fractionalmoodle.tfd215.org/pluginfile.php/64947/mod_resource/content/1/01... · Limits and Continuity . Big Ideas . ... 2008 BC Multiple Choice