Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
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In your post #1 you state that you evaluated the absolute value(s) in the second quote to produce the statement in the third quote. However, I have a counterexample. Set x = a + δ/2 and f(a + δ/2) = L - 2ε. In this case the statement in the second quote is false, whereas the statement in the third quote is true.
Thanks. I love that version of the formal definition!
Sorry that I implied that they themselves are "stand alone" arguments, but in the proof I have seen for the product of the limits, for example, the only appearances of delta is to restate the restriction of the formal definition (the relationship between chosen epsilons and their corresponding deltas), but I think epsilons are just used in the actual arithmetic.
It is correct (but informal) since the following two statements are equivalent.
I've taken all basic Calculus courses and Advanced Calculus I (not that I learned any additional information on proving the basic limit laws in that class...), but I still do not know if what is in my textbook (or what I have researched on the net) is the easiest (intuitive) way to prove the basic limit laws (sum , product, reciprocal, and constant).
Now before I show the arguments, I make note of the formal definition of the (two sided) limit.
I evaluated the absolute value to instead have:
Since this adjusted formal definition shows that the definition can be broken into two parts (we have two statements because of the or operator), we can just handle the first part
, and we can understand that the argument is similar if we were to tackle the second part of the definition.
We can of course prove the difference law with the results of the sum and constant laws, and we can prove the quotient law using the results of the product and reciprocal laws.
Is (*) (the statement in the first quote) a correct assumption? If so, then are these arguments valid proofs?