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## Main Question or Discussion Point

I'm trying to understand [tex]\epsilon-\delta[/tex] proofs, but I'm having some trouble. For example, if we want to prove that [tex]\lim_{x\rightarrow2}x^3=8[/tex], starting from [tex]|x^3-8|[/tex] we get to something like

[tex]|x-2||x^2+2x+4|[/tex]

And this is what confuses me: we conjecture that [tex]|x-2|<1[/tex], then [tex]|x|<3[/tex], so we get

[tex]|x-2||x^2+2x+4| \leq |x-2|||x|^2+2|x|+4|<|x-2||3^2+2*3+4|=19|x-2|[/tex] then we can easily find out what [tex]\delta[/tex] is. But, why do we make that assumption when we're working with every [tex]\epsilon>0[/tex]? That might not be true for some [tex]\epsilon[/tex] ([tex]|x-1|<\delta<1[/tex]. I might be missing something big here, thank you in advance.

[tex]|x-2||x^2+2x+4|[/tex]

And this is what confuses me: we conjecture that [tex]|x-2|<1[/tex], then [tex]|x|<3[/tex], so we get

[tex]|x-2||x^2+2x+4| \leq |x-2|||x|^2+2|x|+4|<|x-2||3^2+2*3+4|=19|x-2|[/tex] then we can easily find out what [tex]\delta[/tex] is. But, why do we make that assumption when we're working with every [tex]\epsilon>0[/tex]? That might not be true for some [tex]\epsilon[/tex] ([tex]|x-1|<\delta<1[/tex]. I might be missing something big here, thank you in advance.