LIMITS OF FUNCTIONS
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
THEOREMS ON LIMITS OF
FUNCTIONS
8) Limit of the nth root of a function
𝑛
lim [ √𝑓(𝑥)] = 𝑛√ lim 𝑓(𝑥)
𝑥→𝑎
𝑥→𝑎
1) Limit of a constant
lim 𝑘 = 𝑘 , where k is constant /
𝑥→𝑎
any real number
2) Limit of the identity function
THEOREMS ON INFINITE LIMITS
𝟏
=
𝒙
𝒙→𝟎 𝒓
𝟏
𝐥𝐢𝐦 =
𝐱→𝟎− 𝐱 𝐫
𝐥𝐢𝐦+
i.
ii.
lim 𝑥 = 𝑎
+∞
+ ∞ 𝐢𝐟 𝐫 𝐢𝐬 𝐞𝐯𝐞𝐧
{
− ∞ 𝐢𝐟 𝐫 𝐢𝐬 𝐨𝐝𝐝
𝑥→𝑎
3) Limit of a constant c times a function
Let lim 𝑓(𝑥) = 𝑐 ≠ 0 and
𝑥→𝑎
lim 𝑔(𝑥) = 0 , then:
lim [𝑐𝑓(𝑥)] = 𝑐 lim 𝑓(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
4) Limit of the sum/difference of two
𝒙→𝒂
functions
If a is positive and g(x) → 0+
lim [𝑓(𝑥) ± 𝑔(𝑥)] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
5) Limit of the product of two
functions
lim [𝑓(𝑥)𝑔(𝑥)] = lim 𝑓(𝑥) lim 𝑔(𝑥)
𝑥→𝑎
𝑥→𝑎
6) Limit of the quotient of two
functions
lim 𝑓(𝑥)
lim [𝑔(𝑥)] = 𝑥→𝑎
, lim 𝑔(𝑥) ≠ 0
lim 𝑔(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
7) Limit of the nth power of a function
𝑛
lim [𝑓(𝑥)]𝑛 = [lim 𝑓(𝑥)] , where n is
𝑥→𝑎
If a is negative and g(x) → 0−
𝒇(𝒙)
2. 𝐥𝐢𝐦 𝒈(𝒙) = −∞
𝒙→𝒂
If a is positive and g(x) → 0−
If a is negative and g(x) → 0+
𝑥→𝑎
𝑓(𝑥)
𝒇(𝒙)
1. 𝐥𝐢𝐦 𝒈(𝒙) = +∞
𝑥→𝑎
any real number
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
Examples:
7) lim
4−𝑥 2
𝑥→2 3−√𝑥 2 +5
=
1) lim1 4𝑥 =
𝑥→
2
𝑥+3
2) lim 𝑥−1 =
𝑥→5
3)
lim (𝑥 + 6)(√𝑥 2 + 3)
𝑥→−1
𝑥 2 −4
𝑥→2 𝑥−2
=
𝑥 3−27
𝑥→3 𝑥 2 −9
=
4) lim
5) lim
6) lim
ℎ→0
(𝑥+ℎ)2 −𝑥 2
ℎ
𝑥 2 +𝑥−2
𝑥→1 (𝑥−1)2
8) lim
=
9) lim
𝑥+1
𝑥→∞ 𝑥+4
=
=
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
3𝑥 3 +5
𝑥→∞ 𝑥 2 +1
10) lim
11) If
3
𝑥 2 +1
𝑓(𝑥+ℎ)−𝑓(𝑥)
lim
ℎ
ℎ→0
12) If
=
𝑓(𝑥) = √4 + 𝑥
𝑓(𝑥+ℎ)−𝑓(𝑥)
lim
ℎ
ℎ→0
,
compute
13) If
𝑓(𝑥) =
𝑓(𝑥) = √1 − 3𝑥
𝑓(𝑥+ℎ)−𝑓(𝑥)
lim
ℎ
ℎ→0
,
compute
,
compute
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
FUNCTIONS
INTERVALS
Interval notation is a way of writing
solutions to algebraic inequalities.
 If 𝑦 = 𝑓(𝑥) =
5𝑥+3
4𝑥−5
, show that 𝑥 = 𝑓(𝑦)
Unbounded Intervals
Compound Inequality
 Show that
Bounded Intervals
𝑓(𝑥+3)
𝑓(𝑥−1)
= 𝑓(4) , if 𝑓(𝑥) = 2𝑥
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022

If 𝑓(𝑥) = 2𝑥 , show that 𝑓(𝑥 + 3) −
𝑓(𝑥 − 1) =
 Compute
15
2
𝑓(𝑥)
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
√3 + 𝑥 , for h = 0
if
𝑓(𝑥) =

If 𝑓(𝑥) =
1+𝑥
1−𝑥
, for h = 0