Originally posted by twhitehead
Let f(x) = 1/x +1
Let L = 0
Let x=1
Let a=2
∀x ∈ ℝ { f(x)≠L } : ∃a ∈ ℝ : a>x ∧ ¬∃ b ∈ ℝ { b>a } : |f(b) − L| ≥ |f(x) − L|
substituting in the final condition
|f(b) − 0| ≥ |2 − 0|
or |f(b)| ≥ |2|
Since f(x) <2 ∀x ∈ ℝ {x>1} we know that ¬∃ b ∈ ℝ { b>a } |f(b)| ≥ |2|
Therefore the claim lim { x→∞ } 1/x +1 = 0 is satisfied by your definition even though we know the true limit is 1.
So now I try;
lim {x→∞} f(x) = L
means;
" there doesn't exist an x that is such that f(x)≠L and x is such that it isn't true that there both exists an a>x such that f(a) is more than twice as close to L as f(x) is to L but there doesn't exist a b>a that is such that f(b) is further away from L than f(a) is from L ".
That can be expressed as:
lim {x→∞} f(x) = L
⇒
¬∃x ∈ ℝ : f(x)≠L ∧ ¬ ( ∃a ∈ ℝ : a>x ∧ 2*|f(a) − L| < |f(x) − L| ∧ ¬∃ b ∈ ℝ : b>a ∧|f(b) − L| > |f(a) − L| )
Hope I got that right at last.
But I find that just a tricky to take in with there being just two many "¬" there so;
PROVIDING f(x) doesn't equal L throughout some infinite/finite non-zero interval of x;
lim x→∞ f(x) = L
means;
for every x where f(x)≠L , there exists an a>x such that f(a) is more than twice as close to L as f(x) is to L but there doesn't exist a b>a that is such that f(b) is further away from L than f(a) is from L.
That can be expressed as:
lim {x→∞} f(x) = L
⇒
∀x ∈ ℝ { f(x)≠L } : ∃a ∈ ℝ : a>x ∧ 2*|f(a) − L| < |f(x) − L| ∧ ¬∃ b ∈ ℝ : b>a ∧|f(b) − L| > |f(a) − L|
if and only if f(x) doesn't equal L throughout some infinite/finite non-zero interval of x.