Math Bridging Course Tutorial 1
Chris TC Wong
16/8/2012 19:00 – 20:00
19/8/2012 16:30 – 17:30
What have we learnt after this two
lectures
• Hard Skills
–
–
–
–
Definition of Limits
Problem Solving Idea : Divide and Conquer
Necessary and Sufficient Condition
Using Maxima as a mathematical tools
• Soft Skills
– How to collaborate with others
– How to stay awake in long lectures
– How to express abstract idea into mathematical
statements
Definition of Limit of Sequence
• ∀𝜖 > 0, ∃𝑁 ∈ ℕ 𝑠. 𝑡. ∀𝑛 > 𝑁, 𝑥𝑛 − 𝐿 < ϵ
Definition of Limit of function
• ∀𝜖 > 0, ∃𝛿 > 0 𝑠. 𝑡. 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 0 < 𝑥 − 𝑐 <
𝛿, 𝑓 𝑥 − 𝐿 < 𝜖
Comparison of two definitions
• ∀𝜖 > 0, ∃𝑁 ∈ ℕ 𝑠. 𝑡. ∀𝑛 > 𝑁, 𝑥𝑛 − 𝐿 < ϵ
• ∀𝜖 > 0, ∃𝛿 > 0 𝑠. 𝑡. 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 0 < 𝑥 − 𝑐 < 𝛿,
𝑓 𝑥 −𝐿 <𝜖
Demonstration to Exercises in Lecture
•
•
1 3
lim
𝑛→∞ 𝑛
3
1
lim
𝑛→∞ 𝑛+1997
6 𝑛
• lim 1 +
𝑛→∞
5𝑛
Demonstration to Exercises in Lecture
•
•
𝑥 2 −1
lim
𝑥→1 𝑥−1
lim 2𝑥
𝑥→0
Necessary and Sufficient Condition
• If A is required to make B happen (no guarantee) , then A is a
necessary condition of B
• E.g.
– You must be registered before being a student in university.
• Registering is a necessary condition for being a student in university.
• Well, you probably need to pay for the registration fee, too.
• If A happen alone can make B happen with guarantee , then A is an
sufficient condition of B
• E.g.
– You must pass AL and scored better than the cut-off score to be an IE
student.
• ‘pass AL and scored better than the cut-off score’ is a sufficient condition for
being an IE student
• But being an IE student do not necessary means that you have passed AL. You
can be a DSE student.
Necessary and Sufficient Condition
(Cont.)
• If A is necessary and sufficient to make B
happens, A is a necessary and sufficient condition
for B.
– E.g. Under normal circumstance, completing circuit
between power button and power supply of a
computer can turn it on.
– E.g. You get a scholarship if and only if you have very
high GPA and is truly outstanding.
– E.g. Love between a couple brings them to marriage.
(?)
If and only if
• If A happens implies B happens, and B
happens implies A happens, then we say A
happens if and only if B happens.
• We says that two things are equivalent under
this condition.
• Some lazy people just write this as “iff”
– E.g. A number can be divided by 9 if and only if
the sum of its digits can be divided by 9.
End