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Mathematics – Grade 12 Notes
Calculus - Limits
Content:
Limits:
1.1 Important Definitions and Notes for Limits
1.2 Important Factorisation Principles for Solving Limits
1.3 Visual Representative Example of What a Limit Is
1.4 Various Worked Examples
1.1 Important Definitions and Notes for Limits
➢ Important notations:
o If a value 𝑥 gets closer and closer to another value 𝑎 then we can symbolise
this the following way:
▪ 𝑥 → 𝑎, which states that 𝑥 tends towards 𝑎 (i.e. 𝑥 gets closer and closer
to 𝑎).
o Using the above point, we can also state that a function tends towards a
specific value:
▪ 𝑓(𝑥) → 𝑏, in this case, the function 𝑓(𝑥) tends towards the value 𝑏 (i.e.
𝑓(𝑥) gets closer and closer to 𝑏).
o By combining these two notations we can state the following:
▪ If 𝑥 → 𝑎 then 𝑓(𝑥) → 𝑏
▪ This statement states that as the value of 𝑥 tends towards 𝑎 the function
𝑓(𝑥) tends towards 𝑏.
o This is where limits come in, we re-write the statement above as:
lim 𝑓(𝑥) = 𝑏
𝑥→𝑎
▪ It states that the limit of a function is the value 𝑏 as 𝑥 gets closer and
closer to 𝑎.
o If 𝑓(𝑥) does not tend to any particular real number (including ∞ and −∞) then
we say that the limit for the function 𝑓(𝑥) does not exist when 𝑥 → 𝑎.
o We assume that limits follow the basic rules and order of solving just like
algebraic expressions (BODMAS, BEDMAS etc.).
1.2 Important Factorisation Principles for Solving Limits
➢ (𝑏 − 𝑎) = −(𝑎 − 𝑏)
➢ Difference of two squares: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
➢ Squaring a difference bracket: (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2
➢ Squaring a sum bracket: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2
Pia Eklund 2023
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➢ Difference of two cubes: 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
➢ Sum of two cubes: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
1.3 Visual Representative Example of What a Limit Is
Consider the following graph:
𝑓(𝑥) = 𝑥 2 + 1
https://www.desmos.com/calculator
Let’s pick an arbitrary value for 𝑥, say 𝑥 = −2:
𝑓(𝑥) = 𝑥 2 + 1
https://www.desmos.com/calculator
Graphically, we can see that when 𝑥 = −2, 𝑦 = 5 which is the same as 𝑓(𝑥) = 5.
Algebraically, if we substitute in the value for 𝑥 = −2 into the function expression 𝑓(𝑥) = 𝑥 2 +
1 we get the same answer as the graphical method. 𝑓(−2) = (−2)2 + 1 = 4 + 1 = 5.
Another way of determining this value is to look at the values to the ‘left’ and ‘right’ of 𝑥 = −2.
Pia Eklund 2023
Mathematics – Grade 12 Notes
Calculus - Limits
Content:
Limits:
1.1 Important Definitions and Notes for Limits
1.2 Important Factorisation Principles for Solving Limits
1.3 Visual Representative Example of What a Limit Is
1.4 Various Worked Examples
1.1 Important Definitions and Notes for Limits
➢ Important notations:
o If a value 𝑥 gets closer and closer to another value 𝑎 then we can symbolise
this the following way:
▪ 𝑥 → 𝑎, which states that 𝑥 tends towards 𝑎 (i.e. 𝑥 gets closer and closer
to 𝑎).
o Using the above point, we can also state that a function tends towards a
specific value:
▪ 𝑓(𝑥) → 𝑏, in this case, the function 𝑓(𝑥) tends towards the value 𝑏 (i.e.
𝑓(𝑥) gets closer and closer to 𝑏).
o By combining these two notations we can state the following:
▪ If 𝑥 → 𝑎 then 𝑓(𝑥) → 𝑏
▪ This statement states that as the value of 𝑥 tends towards 𝑎 the function
𝑓(𝑥) tends towards 𝑏.
o This is where limits come in, we re-write the statement above as:
lim 𝑓(𝑥) = 𝑏
𝑥→𝑎
▪ It states that the limit of a function is the value 𝑏 as 𝑥 gets closer and
closer to 𝑎.
o If 𝑓(𝑥) does not tend to any particular real number (including ∞ and −∞) then
we say that the limit for the function 𝑓(𝑥) does not exist when 𝑥 → 𝑎.
o We assume that limits follow the basic rules and order of solving just like
algebraic expressions (BODMAS, BEDMAS etc.).
1.2 Important Factorisation Principles for Solving Limits
➢ (𝑏 − 𝑎) = −(𝑎 − 𝑏)
➢ Difference of two squares: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
➢ Squaring a difference bracket: (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2
➢ Squaring a sum bracket: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2
Pia Eklund 2023
, 2
➢ Difference of two cubes: 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 )
➢ Sum of two cubes: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 )
1.3 Visual Representative Example of What a Limit Is
Consider the following graph:
𝑓(𝑥) = 𝑥 2 + 1
https://www.desmos.com/calculator
Let’s pick an arbitrary value for 𝑥, say 𝑥 = −2:
𝑓(𝑥) = 𝑥 2 + 1
https://www.desmos.com/calculator
Graphically, we can see that when 𝑥 = −2, 𝑦 = 5 which is the same as 𝑓(𝑥) = 5.
Algebraically, if we substitute in the value for 𝑥 = −2 into the function expression 𝑓(𝑥) = 𝑥 2 +
1 we get the same answer as the graphical method. 𝑓(−2) = (−2)2 + 1 = 4 + 1 = 5.
Another way of determining this value is to look at the values to the ‘left’ and ‘right’ of 𝑥 = −2.
Pia Eklund 2023
