Let f (x) be differentiable and find Lim [f (x + △ x)] ^ 2 - [f (x)] ^ 2 △ x → 0 lim [f(x+△x)]^2-[f(x)]^2 △x→0 =lim{[f(x+△x)+f(x)]*[f(x+△x)-f(x)]}/△x Why is it equal to ＝2f(x)lim[f(x+△x)-f(x)]/△x Especially why is it equal to 2F (x) Please give specific reasons, | Math enthusiast | Mathematical art

Let f (x) be differentiable and find Lim [f (x + △ x)] ^ 2 - [f (x)] ^ 2 △ x → 0 lim [f(x+△x)]^2-[f(x)]^2 △x→0 =lim{[f(x+△x)+f(x)]*[f(x+△x)-f(x)]}/△x Why is it equal to ＝2f(x)lim[f(x+△x)-f(x)]/△x Especially why is it equal to 2F (x) Please give specific reasons,

Because now it's a 0-to-0 limit
We can extract the components which are obviously not equal to 0
Suppose f (x) = 0
Obviously, the proposition holds
So when it's not equal to 0
F (x + △ x) + F (x) can be extracted
Relative to a quantity that is not equal to zero
Delta x is an infinitesimal
It can be ignored
So 2F (x)
I don't know if you understand

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