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lecture 2
Prepared by:
Assist. Lect. Hiba Abdul –Kareem Assist. Lect. Azam Isam
The Limits
Properties of Limits
1. 1-Limit of a constant:
Example :
𝐥𝐢𝐦
𝒙→𝒄
(𝟕)= 7
𝐥𝐢𝐦
𝒙→𝟖
(𝟕)= 7
Properties of Limits
2- Constant Multiplied by a Function (Constant Multiple
Rule)
Example :
lim
𝑥→2
5 𝑥 2=20
Properties of Limits
2- Constant Multiplied by a Function (Constant Multiple
Rule)
Example :
lim
𝑥→2
5 𝑥 2=20
Properties of Limits
3. Sum of Functions
The limit of a sum equals the sum of the limits. In other words,
figure out the limit for each piece, then add them together.
Example :
Properties of Limits
4. Product of Two or More Functions
Example :
Properties of Limits
5. Quotient of Two or More Functions
Example :
Properties of Limits
6.Evaluating the Limit of the Power of a Function
Example :
How to solve limits
How to solve
limits
Multiplying
By Conjugate
Analysis
trigonometric
identities
How to solve limits : Analysis
How to solve limits : Analysis
Example
How to solve limits : Analysis
Example
How to solve limits : Analysis
Example
lim
𝑥→2
𝑥2
− 4
𝑥 − 2
= lim
𝑥→2
𝑥−2 ⋅ 𝑥+2
𝑥−2
=4
Example
lim
𝑥→−2
𝑥3
+ 8
𝑥 + 2
=
lim
𝑥→−2
𝑥+2 𝑥2−2𝑥+4
𝑥+2
=12
How to solve limits : Analysis
Example
How to solve limits : Analysis
Example
How to solve limits : Analysis
Example
Homework
1. lim
𝑥→3
𝑥−3
𝑥2+𝑥−12
2. lim
𝑥→4
8−2𝑥
𝑥2−𝑥−12
How to solve limits : trigonometric
identities
How to solve limits : trigonometric
identities
How to solve limits : trigonometric
identities
Example
lim
𝑥→
𝜋
2
sin 2𝑥
cos − 𝑥
= lim
𝑥→
𝜋
2
2 sin 𝑥 ⋅ cos 𝑥
cos 𝑥
= 2
Example
lim
𝑥→0
sin2
𝑥
1 − cos 2𝑥
= lim
𝑥→0
sin2 𝑥
1 − 1 − 2 sin2 𝑥
= lim
𝑥→0
1
2
= 2
How to solve limits : trigonometric
How to solve limits
• Multiplying By Conjugate
Example
How to solve limits
• Multiplying By Conjugate
Example
L'Hopital's rule
Apply l'Hôpital's rule:
L'Hôpital's rule
Example
L'Hôpital's rule
Example
L'Hôpital's rule
Example
L'Hôpital's rule
Example
L'Hôpital's rule
Homework
lim
𝑥→−2
𝒙 + 𝟏𝟎 + 3𝑥1/3
4𝑥2 + 3𝑥 − 1
Any
Questions

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