As per the definition of limits if <munder> <mo movablelimits="true" form="prefix">lim <m

Joshua Foley 2022-07-01 Answered
As per the definition of limits if lim x a f ( x ) = L, then
ε > 0   δ > 0   s . t 0 < | x a | < δ     0 < | f ( x ) L | < ε
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Answers (2)

poquetahr
Answered 2022-07-02 Author has 18 answers
A different approach for the sake of curiosity.
Let 0 < | x a | < δ ε . Then we have that:
| f ( x ) L | = | x 2 a 2 | = | ( x a ) ( x + a ) | = | x a | | ( x a ) + 2 a | | x a | ( | x a | + 2 | a | ) < δ ε ( δ ε + 2 | a | ) := ε
where you can choose the positive root of the corresponding equation on δ ε

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uplakanimkk
Answered 2022-07-03 Author has 6 answers
Alternative approach:
Without loss of generality, a > 0. That is, the approach for a < 0 is similar, while the approach if a = 0 is trivial.
If δ = ϵ , then | x a | < δ | x a | < ϵ
Instead, take δ = min ( a 2 , ϵ 3 a )
Then, | x a | < δ a 2 < x < 3 a 2
This implies that | x + a | < 5 a 2 < 3 a
So, you have that | x a | < δ and | x + a | < 3 a
Therefore
| x 2 a 2 | = | x a | × | x + a | < δ × 3 a ϵ

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