Finite Limits with Quadratic Factoring Exercises
This collection of four sets of twelve exercises
gives broad practice in finding limits and in factoring quadratics.
Practice in Finding Limits
In each set of twelve exercises, the student practices six types of limits.
The twelve values of a in each set are all different, chosen from the sixteen elements of {± 2, 3, 4, . . . , 9}
A.
a−b
(x − a)(x − b)
=
, a 6= b, a 6= c, b 6= c
x→a (x − a)(x − c)
a−c
lim
Substitution gives 00 , the student factors, and the offending factor cancels.
Exercises #1, 4, 5, 7, 9, 10.
B.
(x − a)2
= 0, a 6= b
x→a (x − a)(x − b)
lim
Substitution gives 00 , the student factors, but the offending factor does not disappear.
Exercise #2.
C.
(x − a)(x − b)
, DNE, a 6= b
x→a
(x − a)2
lim
Substitution gives 00 , the student factors, but the offending factor does not disappear.
Exercise #6.
D.
(a − b)(a − c)
(x − b)(x − c)
=
, a∈
/ {b, c, d, e}, b, c ∈
/ {d, e}, a|bc, a|de
x→a (x − d)(x − e)
(a − d)(a − e)
lim
Substitution gives solution. Many students will factor to no avail, reminding them to substitute first.
I have specified a|bc because otherwise a student could tell just by looking at bc that a is not a root of
the numerator; similarly for a|de.
Exercises #3, 12.
E.
lim
x→a
(x − a)(x − b)
= 0, a, b ∈
/ {c, d}, a|cd
(x − c)(x − d)
Substitution gives
Exercise #8.
F.
lim
x→a
0
1
= 0. Many students will factor to no avail, reminding them to substitute first.
(x − b)(x − c)
, DNE, a ∈
/ {b, c, d}, d ∈
/ {b, c}, a|bc
(x − a)(x − d)
Substitution gives 10 , DNE. Many students will factor to no avail, reminding them to substitute first.
Note that I have excluded the possibility that a = d, which would make the limit infinite. There are no
infinite limits in this collection.
Exercise #11.
Practice in Factoring
All of the quadratics factor over {± 2, 3, 4, . . . , 9}. There are 136 such quadratics, eight of the form
(x − a)(x + a), sixteen (x − a)2 , and 112 (x − a)(x − b), |a| =
6 |b|. The ninety-six quadratics in the forty-eight
exercises are all different. The collection contains all eight of the difference of squares (in exercises 5 and 9)
and twelve of the perfect squares (in exercises 2, 6, and 12). [I couldn’t squeeze in all sixteen.]
In each set of twelve exercises, the twenty-four constant terms are all of different magnitude, chosen
from the thirty-two distince products in [2, 9] × [2, 9], each of which appear at least once in the collection of
four sets.
J. Bradford Burkman
Instructor in Mathematics
Copyright 2010
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Finite Limits with Quadratic Factoring
Set 1
1.
4.
7.
10.
Set 2
1.
4.
7.
10.
Set 3
1.
4.
7.
10.
Set 4
1.
4.
7.
10.
lim
x2 + 17x + 72
x2 + 16x + 63
2.
lim
x2 + 15x + 56
x2 + 14x + 48
5.
lim
x2 + 13x + 42
x2 + 12x + 35
8.
lim
x2 + 15x + 54
x2 + 11x + 30
11.
lim
x2 + 7x − 18
x2 + 6x − 27
2.
lim
x2 + 10x + 16
x2 + 5x − 24
5.
lim
x2 + 9x + 14
x2 + 4x − 21
8.
lim
x2 + 8x + 12
x2 + x − 30
11.
lim
x2 + 6x − 16
x2 + 2x − 48
2.
lim
x2 + 5x − 14
x2 + 2x − 35
5.
lim
x2 + 10x + 24
x2 + 9x + 18
8.
lim
x2 + 3x − 10
x2 + 2x − 15
11.
lim
x2 + 3x − 18
x2 + 2x − 24
2.
lim
x2 − 2x − 35
x2 − 3x − 40
5.
lim
x2 + 7x + 12
x2 + 2x − 8
8.
lim
x2 + 5x + 6
x2 − 4x − 21
11.
x→−9
x→−8
x→−7
x→−6
x→−9
x→−8
x→−7
x→−6
x→−8
x→−7
x→−6
x→−5
x→−6
x→−5
x→−4
x→−3
J. Bradford Burkman
lim
x2 + 6x + 9
x2 + 12x + 27
3.
lim
x2 − 25
+ 14x + 45
6.
x→−3
x→−5 x2
lim
x2 + 11x + 28
x2 + x − 20
9.
lim
x2 − 2x − 15
x2 + 10x + 21
12.
x→4
x→5
lim
x2 + 10x + 25
x2 + 9x + 20
3.
lim
x2 − 9
x2 + 8x + 15
6.
x→−5
x→−3
lim
x2 + 4x − 32
x2 − x − 6
9.
lim
x2 + 3x − 28
x2 + 3x − 40
12.
lim
x2 − 4x + 4
x2 + 4x − 12
3.
x→3
x→4
x→2
lim
x2 − 81
x2 + 5x − 36
6.
lim
x2 − 3x − 54
x2 + 6x + 8
9.
x→−9
x→−2
lim
x2 − 9x + 20
x2 − x − 30
12.
lim
x2 − 8x + 16
x2 − 11x + 28
3.
lim
x2 − 36
x2 − 11x + 30
6.
x2 − 4x − 32
x2 − 3x − 10
9.
x→4
x→4
x→6
lim
x→−2
lim
x→2
x2 − 9x + 14
x2 − 6x − 27
Instructor in Mathematics
Copyright 2010
12.
lim
x→2
x2 + 13x + 40
x2 + 12x + 32
lim
x2 + 11x + 18
x2 + 4x + 4
lim
x2 + 13x + 36
x2 − 16
x→−2
x→−4
lim
x→3
x2 + 18x + 81
x2 + 11x + 24
lim
x→−4
lim
x→6
x2 + 3x − 54
x2 − 12x + 36
lim
x→−2
lim
x→2
x2 + 4x − 45
x2 + 2x − 63
x2 + 7x + 10
x2 − 4
x2 + 16x + 64
x2 + x − 72
lim
x→−4
x2 − x − 72
x2 + x − 42
lim
x2 − 5x + 6
x2 − 6x + 9
lim
x2 + x − 56
x2 − 49
x→3
x→7
lim
x2 − 10x + 25
x2 − 2x − 63
lim
x2 − 2x − 48
x2 − 4x − 45
x→−3
x→−9
lim
x2 − x − 42
x2 − 14x + 49
lim
x2 − x − 56
x2 − 64
x→7
x→8
lim
x→−8
x2 − 18x + 81
x2 − x − 20
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Finite Limits with Quadratic Factoring: Set 1
1.
4.
7.
10.
lim
x2 + 17x + 72
x2 + 16x + 63
2.
lim
x2 + 15x + 56
x2 + 14x + 48
5.
lim
x2 + 13x + 42
x2 + 12x + 35
8.
lim
x2 + 15x + 54
x2 + 11x + 30
11.
x→−9
x→−8
x→−7
x→−6
J. Bradford Burkman
lim
x2 + 6x + 9
x2 + 12x + 27
3.
lim
x2 − 25
+ 14x + 45
6.
x→−3
x→−5 x2
lim
x2 + 11x + 28
x2 + x − 20
9.
lim
x2 − 2x − 15
x2 + 10x + 21
12.
x→4
x→5
Instructor in Mathematics
Copyright 2010
lim
x→2
x2 + 13x + 40
x2 + 12x + 32
lim
x2 + 11x + 18
x2 + 4x + 4
lim
x2 + 13x + 36
x2 − 16
x→−2
x→−4
lim
x→3
x2 + 18x + 81
x2 + 11x + 24
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Finite Limits with Quadratic Factoring: Set 2
1.
4.
7.
10.
lim
x2 + 7x − 18
x2 + 6x − 27
2.
lim
x2 + 10x + 16
x2 + 5x − 24
5.
lim
x2 + 9x + 14
x2 + 4x − 21
8.
lim
x2 + 8x + 12
x2 + x − 30
11.
x→−9
x→−8
x→−7
x→−6
J. Bradford Burkman
lim
x→−5
lim
x2 + 10x + 25
x2 + 9x + 20
x→−3 x2
x2 − 9
+ 8x + 15
3.
6.
lim
x2 + 4x − 32
x2 − x − 6
9.
lim
x2 + 3x − 28
x2 + 3x − 40
12.
x→3
x→4
Instructor in Mathematics
Copyright 2010
lim
x→−4
lim
x→6
lim
x2 + 3x − 54
x2 − 12x + 36
x→−2
lim
x→2
x2 + 4x − 45
x2 + 2x − 63
x2 + 7x + 10
x2 − 4
x2 + 16x + 64
x2 + x − 72
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Finite Limits with Quadratic Factoring: Set 3
1.
4.
7.
10.
lim
x2 + 6x − 16
x2 + 2x − 48
2.
lim
x2 + 5x − 14
x2 + 2x − 35
5.
lim
x2 + 10x + 24
x2 + 9x + 18
8.
lim
x2 + 3x − 10
x2 + 2x − 15
11.
x→−8
x→−7
x→−6
x→−5
J. Bradford Burkman
lim
x→2
x2 − 4x + 4
x2 + 4x − 12
3.
lim
x2 − 81
+ 5x − 36
6.
lim
x2 − 3x − 54
x2 + 6x + 8
9.
x→−9 x2
x→−2
lim
x→4
x2 − 9x + 20
x2 − x − 30
Instructor in Mathematics
Copyright 2010
12.
lim
x→−4
x2 − x − 72
x2 + x − 42
lim
x2 − 5x + 6
x2 − 6x + 9
lim
x2 + x − 56
x2 − 49
x→3
x→7
lim
x→−3
x2 − 10x + 25
x2 − 2x − 63
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Finite Limits with Quadratic Factoring: Set 4
1.
4.
7.
10.
lim
x2 + 3x − 18
x2 + 2x − 24
2.
lim
x2 − 2x − 35
x2 − 3x − 40
5.
lim
x2 + 7x + 12
x2 + 2x − 8
8.
lim
x2 + 5x + 6
x2 − 4x − 21
11.
x→−6
x→−5
x→−4
x→−3
J. Bradford Burkman
lim
x2 − 8x + 16
x2 − 11x + 28
3.
lim
x2 − 36
− 11x + 30
6.
x2 − 4x − 32
x2 − 3x − 10
9.
x→4
x→6 x2
lim
x→−2
lim
x→2
x2 − 9x + 14
x2 − 6x − 27
Instructor in Mathematics
Copyright 2010
12.
lim
x→−9
x2 − 2x − 48
x2 − 4x − 45
lim
x2 − x − 42
x2 − 14x + 49
lim
x2 − x − 56
x2 − 64
x→7
x→8
lim
x→−8
x2 − 18x + 81
x2 − x − 20
Louisiana School for Math, Science, and the Arts
J. Bradford Burkman
[email protected]
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.