Review 1 -_limits-_continuity_(pcalc+_to_ap_calc)

AP*
Calculus
Review
Limits, Continuity, and the
Definition of the Derivative
Teacher Packet
Advanced Placement and AP are registered trademark of the College Entrance Examination Board.
The College Board was not involved in the production of, and does not endorse, this product.
Copyright © 2008 Laying the Foundation, Inc., Dallas, Texas. All rights reserved.
These materials may be used for face-to-face teaching with students only.

Limits, Continuity, and the Definition of the Derivative
Page 1 of 18
DEFINITION Derivative of a Function
The derivative of the function f with respect to the variable x is the function f ′ whose
value at x is
0
( ) (
( ) lim
h
)f x h f x
f x
h→
+ −
′ =
X
Y
(x, f(x))
(x+h, f(x+h))
provided the limit exists.
You will want to recognize this formula (a slope) and know that you need to take the
derivative of ( )f x when you are asked to find
0
( ) (
lim
h
)f x h f x
h→
+ −
.

Page 2 of 18
DEFINITION (ALTERNATE) Derivative at a Point
The derivative of the function f at the point =x a is the limit
( ) ( )
( ) lim
x a
f x f a
f a
x a→
−
′ =
−
X
Y
(x, f(x))
(a, f(a))
provided the limit exists.
This is the slope of a segment connecting two points that are very close together.

Page 3 of 18
DEFINITION Continuity
A function f is continuous at a number a if
1) ( )f a is defined (a is in the domain of f )
2) lim ( )
x a
f x
→
exists
3) lim ( ) ( )
x a
f x f a
→
=
A function is continuous at an x if the function has a value at that x, the function has a
limit at that x, and the value and the limit are the same.
Example:
Given
2
3, 2
( )
3 2,
x x
f x
x x
⎧ + ≤
= ⎨
+ >⎩ 2
Is the function continuous at ?2x =
( ) 7f x =
2
lim ( ) 7
x
f x−
→
= , but the
2
lim ( ) 8
x
f x+
→
=
The function does not have a limit as , therefore the function is not continuous at2x →
2x = .

Page 4 of 18
Limits as x approaches ∞
For rational functions, examine the x with the largest exponent, numerator and
denominator. The x with the largest exponent will carry the weight of the function.
If the x with the largest exponent is in the denominator, the denominator is growing
faster as x → ∞ . Therefore, the limit is 0.
4
3
lim 0
3 7x
x
x x→ ∞
+
=
− +
If the x with the largest exponent is in the numerator, the numerator is growing
faster as x → ∞ . The function behaves like the resulting function when you divide the
x with the largest exponent in the numerator by the x with the largest exponent in the
denominator.
5
2
3
lim
3 7x
x
x x→ ∞
+
= ∞
− +
This function has end behavior like 3
x
5
2
x
x
⎛ ⎞
⎜
⎝ ⎠
⎟ . The function does not reach a limit, but
to say the limit equals infinity gives a very good picture of the behavior.
If the x with the largest exponent is the same, numerator and denominator, the limit
is the coefficients of the two x’s with that largest exponent.
5
5
3 4 4
lim
7 3 7x
x
x x→ ∞
+
=
− + 7
. As x → ∞ , those 5
x terms are like gymnasiums full of sand.
The few grains of sand in the rest of the function do not greatly affect the behavior of the
function as x → ∞ .

Page 5 of 18
LIMITS
lim ( )
x c
f x L
→
=
The limit of f of x as x approaches c equals L.
As x gets closer and closer to some number c (but does not equal c), the value of the
function gets closer and closer (and may equal) some value L.
One-sided Limits
lim ( )
x c
f x L−
→
=
The limit of f of x as x approaches c from the left equals L.
lim ( )
x c
f x L+
→
=
The limit of f of x as x approaches c from the right equals L.
Using the graph above, evaluate the following:
1
lim ( )
x
f x−
→
=
1
lim ( )
x
f x+
→
=
1
lim ( )
x
f x
→
=

Page 6 of 18
Practice Problems
Limit as x approaches infinity
1. 4
3 7
lim
5 8 12x
x
x x→ ∞
−⎛ ⎞
=⎜ ⎟
− +⎝ ⎠
2.
4
4
3 2
lim
5 2 1x
x
x x→ ∞
⎛ ⎞−
=⎜ ⎟
− +⎝ ⎠
3.
6
4
2
lim
10 9 8x
x
x x→ ∞
⎛ ⎞−
=⎜ ⎟
− +⎝ ⎠
4.
4
3 4
7 2
lim
5 2 14x
x
x x→ ∞
⎛ ⎞−
=⎜ ⎟
− −⎝ ⎠
5.
sin
lim xx
x
e→ ∞
⎛ ⎞
=⎜ ⎟
⎝ ⎠
6.
2
9
lim
2 3x
x
x→ −∞
⎛ ⎞−
=⎜ ⎟
⎜ ⎟−⎝ ⎠
7.
2
9
lim
2 3x
x
x→ ∞
⎛ ⎞−
=⎜ ⎟
⎜ ⎟−⎝ ⎠

Page 7 of 18
Practice Problems
Limit as x approaches a number
8. ( )3
2
lim 1
x
x x
→
− +
9.
2
2
4
lim
2x
x
x→
⎛ ⎞−
=⎜ ⎟
−⎝ ⎠
10.
2
3
lim
2x x−
→
⎛ ⎞
=⎜ ⎟
−⎝ ⎠
11.
2
3
lim
2x x+
→
⎛ ⎞
=⎜ ⎟
−⎝ ⎠
12.
2
3
lim
2x x→
⎛ ⎞
=⎜ ⎟
−⎝ ⎠
13.
2
3
lim
2x x+
→
⎛ ⎞
=⎜ ⎟
−⎝ ⎠
14.
4
sin
lim
x
x
xπ
→
⎛ ⎞
=⎜ ⎟
⎝ ⎠
15.
4
tan
lim
x
x
xπ
→
⎛ ⎞
=⎜ ⎟
⎝ ⎠

Page 8 of 18
1. What is
( ) ( )
0
sin sin
lim ?
h
x h x
h→
+ −
(A) sin x (B) cos x (C) sin x−
(D) cos x− (E) The limit does not exist
2.
0
cos cos
3 3
lim
x
x
x
π π
Δ →
⎛ ⎞ ⎛ ⎞
+ Δ −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ =
Δ
(A)
3
2
− (B)
1
2
− (C) 0
(D)
1
2
(E)
3
2
3.
( ) ( )3 3
0
lim
h
x h x
h→
+ −
=
(A) 3
x− (B) 2
3x− (C) 2
3x
(D) 3
x (E) The limit does not exist

Page 9 of 18
4. The graph of is shown above.( )y f x= ( )( )3
2
lim ( ) 3 ( ) 7
x
f x f x
→
− + =
(A) 1 (B) 5 (C) 7 (D) 9 (E) Does not exist
5. If
2
3 4
,
( ) 1
2, 1
x x
x
f x x
x
⎧ − −
≠ −⎪
= +⎨
⎪ = −⎩
1
, what is
1
lim ( )
x
f x
→ −
?
(A) (B) 0 (C) 2 (D) 3 (E) Does not exist5−
6.
6 3
2 6
2 5 10
lim
20 4x
x x
x x→ ∞
⎛ ⎞− +
=⎜ ⎟
− −⎝ ⎠
(A) (B)2−
1
2
− (C)
1
2
(D) 2 (E) Does not exist
7.
5 3
2 6
2 5 10
lim
20 4x
x x
x x→ ∞
⎛ ⎞− +
=⎜ ⎟
− −⎝ ⎠
(A) (B)2−
1
2
− (C) 0 (D)
1
2
(E) 2

Page 10 of 18
8.
1 1
2
lim 1 x
x
e
+
→ ∞
⎛ ⎞
+ =⎜ ⎟
⎝ ⎠
(A) (B) 0 (C)−∞
1
2
e
(D)
1
2
1 e+ (E) ∞
9.
3
5
lim
3x x+
→
=
−
(A) (B) (C) 0−∞ 5−
(D)
5
3
(E) ∞
10. If
3
3
5 1
lim
20 3 2x
n
n kn→ ∞
⎛ ⎞
=⎜ ⎟
− −⎝ ⎠
, then k =
(A) (B) (C)10− 4−
1
4
(D) 4 (E) 10
11. Which of the following is/are true about the function g if
( )
2
2
2
( )
6
x
g x
x x
−
=
+ −
?
I. g is continuous at 2x =
II. The graph of g has a vertical asymptote at 3x = −
III. The graph of g has a horizontal asymptote at 0y =
(A) I only (B) II only (C) III only (D) I and II only (E) II and III only

Page 11 of 18
12.
sin ,
4
( ) cos ,
4
tan ,
4
x x
f x x x
x x
π
π
π
⎧
<⎪
⎪
⎪
= >⎨
⎪
⎪
=⎪⎩
What is
4
lim ( )
x
f xπ
→
?
(A) − (B) 0 (C) 1 (D)∞
2
2
(E) ∞
13. lim
x a
x a
x a→
⎛ ⎞−
=⎜ ⎟
−⎝ ⎠
(A)
1
2 a
(B)
1
a
(C) a (D) 2 a (E) Does not exist
14.
0
ln2
lim
2x
x
x+
→
=
(A) − (B) (C) 0 (D) 1 (E)∞ 1− ∞
15. At , the function given by4x =
2
, 4
( )
4 , 4
x x
h x
x x
⎧ ≤
= ⎨
>⎩
is
(A) Undefined
(B) Continuous but not differentiable
(C) Differentiable but not continuous
(D) Neither continuous nor differentiable
(E) Both continuous and differentiable

Page 12 of 18
Free Response 1
Let h be the function defined by the following:
2
1 3, 1 2
( )
, 2
x x
h x
ax bx x
− + ≤ ≤⎧⎪
= ⎨
− >⎪⎩
a and b are constants.
(a) If and , is continuous for all x in1a = − 4b = − ( )h x [ ]1, ∞ ? Justify your answer.
(b) Describe all values of a and b such that h is a continuous function over the
interval[ ]1, ∞ .
(c) The function h will be continuous and differentiable over the interval [ ]1, ∞ for
which values of a and b?

Page 13 of 18
Free Response 2 (No calculator)
Given the function
3 2
2
2 3
( )
3 3 6
x x x
f x
x x
+ −
=
+ −
.
(a) What are the zeros of ( )f x ?
(b) What are the vertical asymptotes of ( )f x ?
(c) The end behavior model of ( )f x is the function . What is ?( )g x ( )g x
(d) What is lim ( )
x
f x
→ ∞
? What is
( )
lim
( )x
f x
g x→ ∞
?

Page 14 of 18
Key
Page 5: 1, 2, does not exist
Practice Problems:
1. 0
2.
3
5
3. ∞
4.
1
2
−
5. 0
6.
1
2
−
7.
1
2
8. 7
9. 4
10. − ∞
11. ∞
12. does not exist
13. − ∞
14.
2 2
π
15.
4
π

Page 15 of 18
Multiple Choice Questions:
1. B
2. A
3. C
4. B
5. A
6. A
7. C
8. D
9. A
10. A
11. B
12. D
13. A
14. A
15. B

Page 16 of 18
Free Response 1
Let h be the function defined by the following:
2
1 3, 1 2
( )
, 2
x x
h x
ax bx x
⎧ − + ≤ ≤⎪
= ⎨
− >⎪⎩
a and b are constants.
(a) If and , is continuous for all x in1a = − 4b = − ( )h x [ ]1, ∞ ? Justify your answer.
(b) Describe all values of a and b such that h is a continuous function over the interval[ ]1, ∞ .
(c) The function h will be continuous and differentiable over the interval [ ]1, ∞ for which values of a
and b?
(a)
2 1 3 4 2
4 4 2
4 4( 1) 2( 4)
4 4
a b
a b
− + = −
= −
= − − −
=
1 pt equation with substitutions
2 2
lim ( ) lim ( ) 4
(2) 4
x x
f x f x
f
− +
→ →
= =
=
1 pt limits equal; 1 pt value
The function, , is continuous( )h x
for all x, given the a and b,
because the function has a
limit as x approaches 2, the function
has a value as x approaches 2,
and the limit is equal to the value.
(b)
4 2
4 4 2
1
1
2
a b
a
a b
− =
= +
= +
4
b 2 pts for finding a in terms of b
The function is continuous for all
1
1
2
a b= + .
(c)
4 4 2a b= − 1 pt for continuity equation
1 4a b= − 1 pt for differentiability equation
Continued on next page.

Page 17 of 18
3
3
1
2
b
b
a
− =
= −
= −
1 pt for a; 1 pt for b
The function, h(x), is continuous
and differentiable when
1
2
3
a
b
= −
= −

Page 18 of 18
Free Response 2 (No calculator)
Given the function
3 2
2
2 3
( )
3 3 6
x x x
f x
x x
+ −
=
+ −
.
(a) What are the zeros of ( )f x ?
(b) What are the vertical asymptotes of ( )f x ?
(c) The end behavior model of ( )f x is the function ( )g x . What is ( )g x ?
(d) What is lim ( )
x
f x
→ ∞
? What is
( )
lim
( )x
f x
g x→ ∞
?
(a) The zeros of the function, ( )f x , 3 pts, 1 for each zero
occur at 3, 0, 1x = −
(b) There is a vertical asymptote 1 pt for the vertical asymptote
at 2x = −
(c)
1
( )
3
g x x= 2 pts for ( )g x
(d) lim ( )
x
f x
→ ∞
= ∞ 1 pt lim ( )
x
f x
→ ∞
( )
lim 1
( )x
f x
g x→ ∞
= 2 pts for
( )
lim
( )x
f x
g x→ ∞

Review 1 -_limits-_continuity_(pcalc+_to_ap_calc)

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