Comments by "MC116" (@angelmendez-rivera351) on "Proof that f(x) = 1/x is Continuous on (0, infinity) using Delta-Epsilon" video.

  1. I think there is a better proof for the claim that lim 1/x (x —> c) = 1/c for all 0 < c. If 0 < δ & 0 < x & -δ < x – c < δ, then c – δ < x < c + δ. Hence, if you can find δ such that δ < c, then 0 < c – δ, and 1/x < 1/(c – δ, thus 1/(c·x) = 1/|c·x| < 1/[c·(c – δ)], and |x – c|/|c·x| = |1/x – 1/c| < |x – c|/[c·(c – δ)] < δ/[c·(c – δ)]. Therefore, |1/x – 1/c| < δ/[c·(c – δ)]. To prove |1/x – 1/c| < ε, let ε = δ/[c·(c – δ)], equivalent to δ = c^2·ε/(1 + c·ε). Thus, all that remains to be proven is that c^2·ε/(1 + c·ε) < c, in accordance to δ < c, and the proof is complete. c^2·ε/(1 + c·ε) < c is equivalent to c·ε/(1 + c·ε) < 1, equivalent to c·ε < 1 + cε, equivalent to 0 < 1, which is axiomatic. Q. E. D.
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