In trying to prove the limit product rule I've found all explanations(adsbygoogle = window.adsbygoogle || []).push({});

to hit on a point where I lose understanding.

1:If [tex] \lim_{x \to c} f(x) \ = \ L \ and \ \lim_{x \to c} g(x) \ = \ M \ [/tex]

We define the limit as;

[tex] \ \forall \ \epsilon \ >\ 0 \ \exists \ \delta > 0 \ : \ \forall \ x \ \rightarrow \ 0\ < \ | \ x \ - \ c \ | < \delta \ \Rightarrow \ 0 \ < \ | \ f(x)g(x) \ - \ LM \ | \ < \ \epsilon [/tex]

2:Rewrite [tex] f(x) \ = \ L \ + \ (f(x) \ - \ L) \ and \ g(x) \ = \ M \ + \ (g(x) \ - \ M) [/tex]

3:Rewrite [tex] f(x)g(x) \ - \ LM \ as[/tex]

[tex] [L \ + \ (f(x) \ - \ L)] \ [ M \ + \ (g(x) \ - \ M) ] \ - \ LM \ = [/tex]

[tex]LM \ + \ L(g(x) \ - \ M) \ + M(f(x) \ - \ L) \ + \ (f(x) \ - \ L)(g(x) \ - \ M) \ - \ lm [/tex]

[tex] L(g(x) \ - \ M) \ + M(f(x) \ - \ L) \ + \ (f(x) \ - \ L)( g(x) \ - \ M) [/tex]

All this I'm fine with, but next each source I've read confuses me. I'll give the one from Thomas Calculus.

"Since f & g have limits L & M as x-->c, ∃ positive numbers δ_1, δ_2, δ_3, δ_4 such that ∀ x;

[tex]0 \ < \ |x \ - \ c| \ < \delta_1 \Rightarrow \ |f(x) \ - \ L| \ < \ \sqrt{ \frac{ \epsilon }{3} } [/tex]

[tex] 0 \ < \ |x \ - \ c| \ < \delta_2 \Rightarrow \ |g(x) \ - \ M| \ < \ \sqrt{ \frac{ \epsilon }{3} } [/tex]

[tex] 0 \ < \ |x \ - \ c| \ < \delta_3 \Rightarrow \ |f(x) \ - \ L| \ < \ \sqrt{ \frac{ \epsilon }{3(1 \ + \ |M|} } [/tex]

[tex] 0 \ < \ |x \ - \ c| \ < \delta_4 \Rightarrow \ |g(x) \ - \ M| \ < \ \sqrt{ \frac{ \epsilon }{3(1 \ + \ |L|} } [/tex]

What does this even mean and where does it come from???

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# Limit Proof

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