Calculus is study of limit! Proof?
On page 55 at E-Book
But many of us even don’t know what is limit. Well, limit is a tool to approximate value, let say you have 𝑥 = 5 and you say 5 𝑥 = 5 5 = 1 this is true for 𝑥 = 5. But, when you use limit we tend to say lim 𝑥→5 5 𝑥 = lim 𝑥→5 5 5 = 1 means limit as x approaches to 5 of 5 𝑥 approaches 1.
So even thou it’s looks obvious, limits are used to approximate values. Maybe this is kind a boring but when you approach something like 0 things getting interesting~
Let say you have lim 𝑥→ 0 𝑦/𝑥 , 𝑦 ∈ ℝ. The value of that limit where x approaches 0 is infinity.
Maybe at a glance it doesn’t make sense but when we think about the approaches meaning it make sense.
x = 0.1, 𝑦/𝑥 = 10𝑦
x = 0.01, 𝑦/𝑥 = 100𝑦
x = 0.001, 𝑦/𝑥 = 100𝑦
x = 0.000…0001, 𝑦/𝑥 = 1000…000𝑦 = ∞,
this is logic, when x tends to be zero~ the limit of 𝑦/𝑥 = ∞.
Now I hope you understand how limit works because that’s not the main point we want to explain in here.
Do you know who is the father of calculus? Clue: Newton
He invented calculus when he is 25 years old, just amazing how back then they discovery such beauty in this life
He define the fundamental of calculus which is rate of change
𝑑𝑦/𝑑𝑡 = 𝑓′(𝑥) = lim 𝑡→0 𝑓(𝑥 + 𝑡) − 𝑓(𝑥) 𝑡
First, 𝑑𝑦/𝑑𝑡 means rate of change to t (time). Usually t denotes time in math.
So the logic is, rate of change by a little (1) time subtracted by it’s previous value and (2) divided with time that elapsed is the rate of change by time.
(1) 𝑓(𝑥 + 𝑡) − 𝑓(𝑥)
(2) lim 𝑡→0 𝑓(𝑥+𝑡)−𝑓(𝑥) 𝑡
So that the famous rate of change, it may look ugly because we all know the formula, but as a mathmagician you should know this knowledge to tell your friend right?
Here I list some useful formula if you learn calculus 😊
Have a good day. 
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