Lecture Notes in Calculus
Page 1 of 2
Prepared by: Prof. Sarah Jean Q. Cabanig
•
Intuitive Idea of the Limit of a Function
Consider 𝑓(𝑥) = 3 − 𝑥 2 . Observe the behavior of the values of 𝑓(𝑥) as 𝑥 approaches 2.
𝒙
1.99
1.999
1.9999
1.99999
2
2.00001
2.0001
2.001
2.01
𝒇(𝒙) = 𝟑 − 𝒙𝟐
▪
Definition: Limit of a Function [Intuitive]
Let 𝑓 be a function defined in an open interval 𝑐, except possibly at 𝑎. The limit 𝐿 of a function
𝑓(𝑥) as 𝑥 approaches 𝑎 means that the values of 𝑓(𝑥) get closer and closer to 𝐿 as 𝑥 assumes
values going closer and closer (but not reaching) to 𝑎.
In symbol we write,
lim 𝑓(𝑥) = 𝐿
𝑥→𝑎
▪
Illustrations:
1.
Now consider 𝑓(𝑥) = 𝑥 + 1, 𝑔(𝑥) =
𝒙
𝒇(𝒙) = 𝒙 + 𝟏
𝒈(𝒙) =
𝒙𝟐 − 𝟏
𝒙−𝟏
𝒙 + 𝟏 if 𝒙 ≠ 𝟏
𝒉(𝒙) = {
𝟏
if 𝒙 = 𝟏
𝑥 + 1 if 𝑥 ≠ 1
𝑥2 − 1
, and ℎ(𝑥) = {
.
𝑥−1
1
if 𝑥 = 1
0.99
0.999
0.9999
1
1.0001
1.001
1.01
Lecture Notes in Calculus
Page 2 of 2
Prepared by: Prof. Sarah Jean Q. Cabanig
2.
1 if 𝑥 > 0
Consider the 𝑠𝑔𝑛(𝑥) = { 0 if 𝑥 = 0
−1 if 𝑥 < 0
If 0 < 𝑥 < 0.01, then 𝑠𝑔𝑛(𝑥) = 1. While if −0.01 < 𝑥 < 0, then 𝑠𝑔𝑛(𝑥) = −1. Hence, 𝑠𝑔𝑛(𝑥)
does not approach a single value as 𝑥 approaches 0. Therefore, lim 𝑠𝑔𝑛 (𝑥) does not exist.
𝑥→0
3.
Let ℎ(𝑥) =
1
, observe the values of ℎ(𝑥) as 𝑥 approaches 0.
𝑥2
Since the values of ℎ(𝑥) do not tend to any “finite” number as 𝑥 approaches to 0, then
1
does not exist.
𝑥→0 𝑥 2
lim
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Limit Theorems
1. If the limit of a function exists then it must me unique.
2. If 𝑐 ∈ ℝ, then lim 𝑐 = 𝑐.
𝑥→𝑎
3. lim 𝑥 = 𝑎.
𝑥→𝑎
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Operations on Limits
Let 𝑓 and 𝑔 be functions, 𝑎 and 𝑐 be real numbers, and 𝑛 be an integer.
1.
2.
3.
4.
lim [𝑓(𝑥) ± 𝑔(𝑥)] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
lim 𝑐 ∙ [𝑓(𝑥)] = 𝑐 [lim 𝑓(𝑥)]
𝑥→𝑎
𝑥→𝑎
lim [𝑓(𝑥)][𝑔(𝑥)] = [lim 𝑓(𝑥)] [lim 𝑔(𝑥)]
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
lim 𝑓(𝑥)
𝑓(𝑥) 𝑥→𝑎
=
; lim 𝑔(𝑥) ≠ 0
𝑥→𝑎 𝑔(𝑥)
lim 𝑔(𝑥) 𝑥→𝑎
lim
𝑥→𝑎
5.
6.
lim [𝑓(𝑥)]𝑛 = [lim 𝑓(𝑥)]
𝑥→𝑎
𝑛
𝑥→𝑎
𝑛
lim √𝑓(𝑥) = 𝑛√ lim 𝑓(𝑥) ; for all lim 𝑓(𝑥) ≥ 0 when 𝑛 is even.
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎