Limit Theorems

1. LIMITS
Calculus

2. DEFINITI0N OF THE LIMIT OF A FUNCTION
• Let 𝑓 be a function defined at every number
in some open interval containing 𝑎, except
possibly at the number itself. The limit of 𝒇(𝒙)
as x approaches a is L, written as
lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
if the following statement is true:
Given any 𝜀 > 0, however small, there
exist a 𝛿 > 0 such that if 0 < 𝑥 − 𝑎 < 𝛿 then
𝑓 𝑥 − 𝐿 < 𝜀
𝐿 + 𝜀1
𝐿 − 𝜀1
𝐿
𝛼 − 𝛿1 𝛼 + 𝛿1
𝛼

3. LIMIT THEOREMS
• Limit Theorem 1 (Limit of a constant)
• Let c be a constant, for any number a
lim
𝑥→𝑎
𝑐 = 𝑐
Illustration:
lim
𝑥→8
29 = 29
• Limit Theorem 2 (Limit of Identity
Function)
lim
𝑥→𝑎
𝑥 = 𝑎
Illustration:
lim
𝑥→12
𝑥 = 12
• Limit Theorem 3 (Limit of Linear
Function)
• Let m and b be any constant
lim
𝑥→𝑎
𝑚𝑥 + 𝑏 = 𝑚𝑎 + 𝑏
Illustration:
lim
𝑥→5
5𝑥 + 30 = 5 5 + 30 = 55
• Limit Theorem 4 ( Sum and Difference
of Two Functions)
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝐾, then
lim
𝑥→𝑎
𝑓 𝑥 ± 𝑔 𝑥 = 𝐿 ± 𝐾
Illustration:
lim
𝑥→3
𝑥2
+ 2𝑥 = 32
+ 2 3 = 15

4. LIMIT THEOREMS
• Limit Theorem 5 (Product of functions)
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝐾, then
lim
𝑥→𝑎
𝑓 𝑥 ∗ 𝑔 𝑥 = 𝐿 ∗ 𝐾
Illustration:
lim
𝑥→3
(𝑥2
)(2𝑥) = 32
2 ∗ 3 = 54
• Limit Theorem 6 (𝑛𝑡ℎ
Power of Function)
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and n is any positive integer,
then
lim
𝑥→𝑎
𝑓 𝑥 𝑛 = 𝐿𝑛
Illustration:
lim
𝑥→−2
5𝑥 + 3 2
= lim
𝑥→−2
5𝑥 + 3
2
= −7 2
= 49
• Limit Theorem 7 (Quotient of Functions)
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝐾, then
lim
𝑥→𝑎
𝑓 𝑥
𝑔 𝑥
=
𝐿
𝐾
; 𝐾 ≠ 0
Illustration:
lim
𝑥→−1
𝑥
−3𝑥+5
=
lim
𝑥→−1
𝑥
lim
𝑥→−1
−3𝑥+5
=
−1
8
• Limit Theorem 8 (𝑛𝑡ℎ
Root of Function)
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and n is any positive integer,
then
lim
𝑥→𝑎
𝑛
𝑓 𝑥 =
𝑛
𝐿
Illustration:
lim
𝑥→−1
3 𝑥
−3𝑥 + 5
=
3
lim
𝑥→−1
𝑥
−3𝑥 + 5
=
3 −1
8
=
−1
2

5. LIMIT THEOREMS
• Limit Theorem 9
If a is any real number except zero, then
lim
𝑥→𝑎
1
𝑥
=
1
𝑎
Illustration:
lim
𝑥→−13
1
𝑥
=
−1
13
• Limit Theorem 10
If 𝑎 > 0 and n is a positive integer, or if 𝑎 ≤
0 and n is an odd positive integer,
then
lim
𝑥→𝑎
𝑛
𝑥 = 𝑛
𝑎
Illustration:
lim
𝑥→−3
5
81𝑥 = 5
lim
𝑥→−3
81𝑥 =
5
81 −3 =
5
−243 = −3
• Limit Theorem 11
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 if and only if lim
𝑥→𝑎
(
)
𝑓 𝑥 −
𝐿 = 0
• Limit Theorem 12
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 if and only if lim
𝑥→𝑎
𝑡 + 𝑎 = 𝐿
• Limit Theorem 13
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿1 and lim
𝑥→𝑎
𝑓 𝑥 = 𝐿2 then
𝐿1 = 𝐿2

6. EXERCISES
Find the limit of each of the following.
• lim
𝑥→3
5𝑥 + 30 =
• lim
𝑥→−8
𝑥2
+ 5𝑥 + 30 =
• lim
𝑥→5
𝑥+12
13𝑥+14
=
• lim
𝑥→−1
2𝑥+1
𝑥2−3𝑥−4
=
• lim
𝑥→2
𝑥3+3𝑥+5
𝑥4−1
=
• lim
𝑥→3
3 5+2𝑥
5−𝑥
=

7. lim
𝑥→−1
2𝑥 + 1
𝑥2 − 3𝑥 − 4
=

8. EXERCISES
• lim
𝑘→3
𝑘2−6𝑘+9
𝑘−3
• lim
𝑙→ Τ
1
3
3𝑙−1
9𝑙2−1
• lim
𝑚→4
3𝑚2−17𝑚+20
4𝑚2−25𝑚+36
=

9. EXERCISES
• lim
𝑠→−1
𝑠3+1
𝑠+1
=
• lim
𝑥→ Τ
3
2
8𝑥3−27
4𝑥2−9
=
• lim
𝑠→−1
𝑠+5−2
𝑠+1
=

10. Table of Values
𝑦 =
1
𝑥

11. lim
𝑥−3
𝑥2
− 6𝑥 + 9
𝑥 − 3

14. EXERCISES
• lim
𝑘→1
3
𝑘−1
𝑘−1
• lim
𝑚→−1
2𝑚2−𝑚−3
𝑚3+2𝑚2+6𝑚+2
=
• lim
𝑠→−2
𝑠3−𝑠2−𝑠+10
𝑠2+3𝑠+2
=
• lim
𝑥→0
𝑥
𝑥+1−1
=

15. ONE-SIDED LIMITS
• Right-Hand Limit
• Let 𝑓 be a function defined at
every number in some open
interval 𝑎, 𝑐 . Then the limit of
𝑓(𝑥), x approaches a from the
right, is L, written
lim
𝑥→𝑎+
𝑓(𝑥) = 𝐿
if for any 𝜀 > 0, however small,
there exist a 𝛿 > 0 such that if
0 < 𝑥 − 𝑎 < 𝛿 then
𝑓 𝑥 − 𝐿 < 𝜀.
• Left-Hand Limit
• Let 𝑓 be a function defined at
every number in some open
interval 𝑑, 𝑎 . Then the limit of
𝑓(𝑥), x approaches a from the
left, is L, written
lim
𝑥→𝑎−
𝑓(𝑥) = 𝐿
if for any 𝜀 > 0, however small,
there exist a 𝛿 > 0 such that if
0 < 𝑎 − 𝑥 < 𝛿 then
𝑓 𝑥 − 𝐿 < 𝜀.

16. ONE-SIDED LIMITS
Theorem 14
• The lim
𝑥→𝑎
𝑓(𝑥) exist and is equal to L if
and only if lim
𝑥→𝑎+
𝑓(𝑥) and lim
𝑥→𝑎−
𝑓(𝑥) both
exist and is both equal to L.
Examples
• Given the cost function
𝐶 𝑥 = ቊ
2𝑥 𝑖𝑓 0 ≤ 𝑥 ≤ 10
1.8𝑥 𝑖𝑓 10 < 𝑥
, find the
limits as x approaches 10.

17. EXAMPLES
• Let h be defined by
ℎ 𝑥 = ቊ
𝑥 𝑖𝑓 𝑥 ≠ 0
2 𝑖𝑓 𝑥 = 0
find the limit of h as x approaches 0 if it
exist.
• Let f be defined by
𝑓 𝑥 = ൝
4 − 𝑥2
𝑖𝑓 𝑥 ≤ 1
2 + 𝑥2
𝑖𝑓 𝑥 > 1
find the limit of f as x approaches 1 if
it exist.

18. EXAMPLES
• Let 𝑙 be defined by
𝑙 𝑥 =
𝑥 − 3
𝑥 − 3
Find the limit of h as x approaches 3, if it
exist.
• Let k be defined by
𝑘 𝑥 = ൞
𝑥 + 5 𝑖𝑓 𝑥 < −3
9 − 𝑥2 𝑖𝑓 − 3 ≤ 𝑥 ≤ 3
3 − 𝑥 𝑖𝑓 3 < 𝑥
find the limit of k as x approaches 3
and - 3, if it exist.

19. INFINITE LIMITS
• Definition of Function Values
Increasing Without Bound
• Let 𝑓 be a function defined at
every number in some open
interval 𝐼 containing a, except
probably at the number a itself.
As x approaches a, 𝑓(𝑥)
increases without bound which
is written
lim
𝑥→𝑎
𝑓(𝑥) = +∞
if for any 𝑁 > 0 there exist a
𝛿 > 0 such that if 0 < 𝑥 − 𝑎 < 𝛿
then 𝑓 𝑥 > 𝑁.
• Definition of Function Values
Decreasing Without Bound
• Let 𝑓 be a function defined at
every number in some open
interval 𝐼 containing a, except
probably at the number a itself.
As x approaches a, 𝑓(𝑥)
decreases without bound
which is written
lim
𝑥→𝑎
𝑓(𝑥) = −∞
if for any 𝑁 < 0 there exist a
𝛿 > 0 such that if 0 < 𝑥 − 𝑎 < 𝛿
then 𝑓 𝑥 < 𝑁.

20. LIMIT THEOREMS
• Limit Theorem 15
If r is any positive integer, then
𝑖) lim
𝑥→0+
1
𝑥𝑟
= +∞
𝑖𝑖) lim
𝑥→0−
1
𝑥𝑟
= ቊ
−∞ 𝑖𝑓 𝑟 𝑖𝑠 𝑜𝑑𝑑
+∞ 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛
Illustration:
• lim
𝑥→0+
1
𝑥5 = +∞
• lim
𝑥→0+
1
𝑥4 = +∞
• lim
𝑥→0−
1
𝑥5 = −∞
• lim
𝑥→0−
1
𝑥4 = +∞
• Limit Theorem 16
If a is any real number and if lim
𝑥→𝑎
𝑓 𝑥 = 0
and if the lim
𝑥→𝑎
𝑔 𝑥 = 𝑐, where c is a
constant not equal to zero, then
i. If 𝑐 > 0 and if 𝑓 𝑥 → 0 through
positive values of 𝑓 𝑥 ,
lim
𝑥→𝑎
𝑔(𝑥)
𝑓(𝑥)
= +∞
ii. If 𝑐 > 0 and if 𝑓 𝑥 → 0 through
negative values of 𝑓 𝑥 ,
lim
𝑥→𝑎
𝑔(𝑥)
𝑓(𝑥)
= −∞

21. LIMIT THEOREMS
iii. If 𝑐 < 0 and if 𝑓 𝑥 → 0 through
positive values of 𝑓 𝑥 ,
lim
𝑥→𝑎
𝑔(𝑥)
𝑓(𝑥)
= −∞
iv. If 𝑐 < 0 and if 𝑓 𝑥 → 0 through
negative values of 𝑓 𝑥 ,
lim
𝑥→𝑎
𝑔(𝑥)
𝑓(𝑥)
= +∞
The theorem is also valid for sided
limits.
• Illustrations
• lim
𝑥→1−
2𝑥
𝑥−1
=
• lim
𝑚→3+
𝑚2+𝑚+2
𝑚2−2𝑚−3
=
• lim
𝑚→−3+
𝑚2+𝑚+2
𝑚2−2𝑚−3
=

22. LIMIT THEOREMS
• Limit Theorem 17
• If lim
𝑥→𝑎
𝑓 𝑥 = +∞ and the lim
𝑥→𝑎
𝑔 𝑥 = 𝑐,
where c is any constant, then
lim
𝑥→𝑎
𝑓 𝑥 + 𝑔(𝑥) = +∞
• If lim
𝑥→𝑎
𝑓 𝑥 = −∞ and the lim
𝑥→𝑎
𝑔 𝑥 = 𝑐,
where c is any constant, then
lim
𝑥→𝑎
𝑓 𝑥 + 𝑔(𝑥) = −∞
• Limit Theorem 18
If lim
𝑥→𝑎
𝑓 𝑥 = +∞ and the lim
𝑥→𝑎
𝑔 𝑥 = 𝑐, where
c is any constant except 0, then
i. If c > 0, lim
𝑥→𝑎
𝑓 𝑥 ∗ 𝑔(𝑥) = +∞
ii. If c < 0, lim
𝑥→𝑎
𝑓 𝑥 ∗ 𝑔(𝑥) = −∞

23. LIMIT THEOREMS
• Limit Theorem 19
If lim
𝑥→𝑎
𝑓 𝑥 = −∞ and the lim
𝑥→𝑎
𝑔 𝑥 = 𝑐, where
c is any constant except 0, then
i. If c > 0, lim
𝑥→𝑎
𝑓 𝑥 ∗ 𝑔(𝑥) = −∞
ii. If c < 0, lim
𝑥→𝑎
𝑓 𝑥 ∗ 𝑔(𝑥) = +∞
• Definition of Vertical Asymptote
• The line 𝑥 = 𝑎 is a vertical
asymptote of the graph of the
function 𝑓 if at least one of the
following statements is true:
i. lim
𝑥→𝑎+
𝑓 𝑥 = +∞
ii. lim
𝑥→𝑎+
𝑓 𝑥 = −∞
iii. lim
𝑥→𝑎−
𝑓 𝑥 = +∞
iv. lim
𝑥→𝑎−
𝑓 𝑥 = −∞

24. EXERCISES
• Find the limit analytically
• lim
𝑥→2+
𝑥+2
𝑥2−4
=
• lim
𝑥→2−
𝑥+2
𝑥2−4
=
• lim
𝑥→2
𝑥+2
𝑥2−4
=
• Find the limit analytically
• lim
𝑥→0−
3+𝑥2
𝑥
=
• lim
𝑥→3+
𝑥2−9
𝑥−3
=
• lim
𝑥→0+
1
𝑥
−
1
𝑥2 =

25. • Find the limit analytically
• lim
𝑥→0−
2 − 4𝑥3
5𝑥2+3𝑥3 =
• lim
𝑥→2−
1
𝑥−2
−
3
𝑥2−4
=
• lim
𝑠→−4−
3
𝑠2+3𝑠−4
−
5
𝑠+4
=
• Find the limit analytically
• lim
𝑥→3−
𝑥3+9𝑥2+20𝑥
𝑥2+𝑥 −12
=
• lim
𝑥→1+
𝑥−1
2𝑥−𝑥2−1
=
• lim
𝑥→2−
𝑥−3
2− 4𝑥−𝑥2
=

26. LIMIT THEOREMS
• Limit Theorem 20 (Limit of a Composite
Function)
If lim
𝑥→𝑎
𝑔 𝑥 = 𝑏 and if the function 𝑓 is
continuous at b, then
lim
𝑥→𝑎
𝑓 ∘ 𝑔 𝑥 = 𝑏 or, equivalently,
lim
𝑥→𝑎
𝑓 𝑔 𝑥 = 𝑓 lim
𝑥→𝑎
𝑔 𝑥 .
• Squeeze Theorem
• Suppose that the function 𝑓, 𝑔, and ℎ
are defined on some open interval 𝐼
containing 𝑎 except possibly at 𝑎 itself,
and that 𝑓 𝑥 ≤ 𝑔(𝑥) ≤ ℎ(𝑥) for all 𝑥 in 𝐼
for which 𝑥 ≠ 𝑎. Also suppose that
lim
𝑥→𝑎
𝑓 𝑥 and lim
𝑥→𝑎
ℎ 𝑥 both exist and
are equal to 𝐿. Then lim
𝑥→𝑎
𝑔 𝑥 exist and
is equal to 𝐿.

27. LIMIT THEOREMS
• Limit Theorem 21
lim
𝑚→0
sin 𝑚
𝑚
= 1
• Limit Theorem 22
lim
𝑚→0
1 − cos 𝑚
𝑚
= 0
• Example:
• lim
𝑥→0
1 − cos 𝑥
sin 𝑥
=
• lim
𝑥→0
2 𝑡𝑎𝑛2𝑥
𝑥2 =
• lim
𝑥→0
sin 4𝑥
𝑥
=
• lim
𝑥→0
3𝑥2
1 −𝑐𝑜𝑠2 1
2
𝑥
=

29. LIMITS AT INFINITY
• Definition of the limit of 𝑓(𝑥) as x
Increases without Bound
• Let 𝑓 be a function that is defined at
every number in some interval 𝑎, +∞ .
The limit of 𝑓 𝑥 , as x increases without
bound, is 𝐿, written lim
𝑥→+∞
𝑓 𝑥 = 𝐿 if for
any 𝜀 > 0, however small, there exist a
number 𝑁 > 0 such that if 𝑥 > 𝑁 then
𝑓 𝑥 − 𝐿 < 𝜀.
• Definition of the limit of 𝑓(𝑥) as x
Decreases without Bound
• Let 𝑓 be a function that is defined at
every number in some interval −∞, 𝑎 .
The limit of 𝑓 𝑥 , as x decreases
without bound, is 𝐾, written lim
𝑥→−∞
𝑓 𝑥 =
𝐾 if for any 𝜀 > 0, however small, there
exist a number 𝑁 > 0 such that if 𝑥 > 𝑁
then 𝑓 𝑥 − 𝐾 < 𝜀.

30. LIMIT THEOREM
• Limit Theorem 23
• If r is any positive integer, then
i. lim
𝑥→+∞
1
𝑥𝑟 = 0
ii. lim
𝑥→−∞
1
𝑥𝑟 = 0
• Illustration
• lim
𝑥→+∞
1
𝑥5 =
• lim
𝑥→−∞
1
𝑥6 =
• lim
𝑥→+∞
𝑥4 − 1
𝑥5 + 3
=
• lim
𝑥→−∞
𝑥 − 1
𝑥 + 3
=
• lim
𝑥→+∞
4𝑥5 −3𝑥 + 1
2𝑥5 + 3
=
• lim
𝑥→+∞
3𝑥 − 4
4𝑥2 − 8𝑥 + 4
=
• lim
𝑥→+∞
𝑥2
𝑥 − 21
=

31. LIMIT THEOREM
• Definition of Horizontal Asymptote
• The line 𝑦 = 𝑏 is a horizontal asymptote
of the graph of the function 𝑓 if at least
one of the statement is true:
• lim
𝑥→+∞
𝑓 𝑥 = 𝑏, and for some number N,
if 𝑥 > 𝑁, then 𝑓 𝑥 ≠ 𝑏.
• lim
𝑥→−∞
𝑓 𝑥 = 𝑏, and for some number N,
if 𝑥 < 𝑁, then 𝑓 𝑥 ≠ 𝑏
• Find the horizontal asymptote of 𝑓 𝑥 =
𝑥
𝑥2 − 1
• Limit Theorem 24
• lim
𝑥→±∞
𝑥𝑛
= ∞ when 𝑛 is even
• lim
𝑥→∞
𝑥𝑛
= ∞ and lim
𝑥→−∞
𝑥𝑛
= −∞ when 𝑛
is odd.
• Limit Theorem 25
• lim
𝑥→±∞
𝑝(𝑥) = lim
𝑥→±∞
𝑎𝑛𝑥𝑛 = ∞ or −∞,
depending on the degree of the
polynomial and the sign of the leading
coefficient 𝑎𝑛.

32. LIMIT THEOREM
• End Behavior and Asymptotes of
Rational Functions
• Suppose 𝑓 𝑥 =
𝑝(𝑥)
𝑞(𝑥)
is a rational
function, where
𝑝 𝑥 = 𝑎𝑚𝑥𝑚
+ 𝑎𝑚−1𝑥𝑚−1
+ ⋯ +
𝑎2𝑥2
+ 𝑎1𝑥 + 𝑎0 and
𝑞 𝑥 = 𝑏𝑚𝑥𝑚
+ 𝑏𝑚−1𝑥𝑚−1
+ ⋯ +
𝑏2𝑥2
+ 𝑏1𝑥 + 𝑏0,
with 𝑎𝑛 ≠ 0 and 𝑏𝑛 ≠ 0.
• Degree of numerator less than degree
of denominator
If 𝑚 < 𝑛, then lim
𝑥→±∞
𝑓(𝑥) = 0, and 𝑦 = 0 is
a horizontal asymptote of 𝑓.
• Degree of numerator equals degree
of denominator
If 𝑚 = 𝑛, then lim
𝑥→±∞
𝑓(𝑥) =
𝑎𝑚
𝑏𝑛
, and 𝑦 =
𝑎𝑚
𝑏𝑛
is a horizontal asymptote of 𝑓.

33. LIMIT THEOREM
• Degree of numerator greater
than degree of denominator
If 𝑚 > 𝑛, then lim
𝑥→±∞
𝑓(𝑥) = ∞ or
− ∞, and 𝑓 has no horizontal
asymptote.
• Assuming that 𝑓(𝑥) is in
reduced form (𝑝 and 𝑞 share
no common factors), vertical
asymptotes occur at the zeros
of 𝑞.
• End Behavior of ℯ𝑥
, ℯ−𝑥
and ln 𝑥
• End Behavior of ℯ𝑥
= ∞ and ℯ−𝑥
on −∞, ∞ and ln 𝑥 on 0, ∞ is
given by the following limits:
• lim
𝑥→∞
ℯ𝑥
= ∞ and lim
𝑥→−∞
ℯ𝑥
= 0,
• lim
𝑥→∞
ℯ−𝑥
= 0 and lim
𝑥→−∞
ℯ−𝑥
= ∞,
• lim
𝑥→0+
ln 𝑥 = −∞ and lim
𝑥→∞
ln 𝑥 = ∞.

34. EXERCISES
Find the limit of each of the following.
• lim
𝑥→3
5𝑥 + 30 =
• lim
𝑥→−8
𝑥2
+ 5𝑥 + 30 =
• lim
𝑥→5
𝑥+12
13𝑥+14
=
• lim
𝑥→−1
2𝑥+1
𝑥2−3𝑥−4
=
• lim
𝑥→2
𝑥3+3𝑥+5
𝑥4−1
=
• lim
𝑥→3
3 5+2𝑥
5−𝑥
=

35. EXERCISE
Find the limit if it exist.
• lim
𝑥→5
12𝑥 + 13 =
• lim
𝑥→5
𝑥 −5
𝑥2 + 25
=
• lim
𝑥→−1
൞
𝑥 − 1 𝑖𝑓 𝑥 < −1
𝑜 𝑖𝑓 𝑥 = 1
1 − 𝑥 𝑖𝑓 − 1 < 𝑥
• Given 𝑓 𝑥 =
ቊ
3𝑥 + 2, 𝑖𝑓 𝑥 < 4
5𝑥 + 𝑘, 𝑖𝑓 4 ≤ 𝑥
, find the
value of k such that
lim
𝑥→4
𝑓(𝑥) exist
Show your solution.
• lim
𝑚→12
5𝑚 + 30 =
• lim
𝑚→−3
15 + 5𝑥
𝑥2 + 9
=
• lim
𝑚→−3
൞
𝑚2 − 9 𝑖𝑓 𝑚 ≤ −3
9 − 𝑚2 𝑖𝑓 − 3 < 𝑚 < 3
𝑚2 − 9 𝑖𝑓 3 ≤ 𝑚
• Given 𝑓 𝑚 =
ቊ
𝑘𝑚 − 3, 𝑖𝑓 𝑚 ≤ −1
𝑚2
+ 𝑘, 𝑖𝑓 − 1 < 𝑥
, find the
value of k such that
lim
𝑚→−1
𝑓(𝑚) exist.
Box your answers.
• lim
𝑙→−3
𝑙 − 29 =
• lim
𝑙→−12
𝑙
2
−12
144 +𝑙2 =
• lim
𝑙→0
൝
3
𝑙 𝑖𝑓 𝑙 < 0
𝑙 𝑖𝑓 𝑙 ≥ 0
• Given 𝑓 𝑙 =
൞
𝑙2 + 2, 𝑖𝑓 𝑙 ≤ −2
𝑎𝑙 + 𝑏, 𝑖𝑓 − 2 < 𝑙 < 2
2𝑙 − 6, 𝑖𝑓 2 ≤ 𝑙
,
find the value of a and b
such that lim
𝑙→−2
𝑓(𝑙) exist.