Why does the book say if f(x) is continuous at a then(adsbygoogle = window.adsbygoogle || []).push({});

lim ( f(a + Δx) - f(a) ) / Δx, that Δx will go to zero. What does that mean geometrically?

Δx→0

More importantly, why would Δx not approach zero if f(x) is not continuous at a?

Im guessing it has something to do with the slopes of the secant lines approaching a definite number therefore Δx will approach 0. However say we were given g(x) = 1/(x - a)

Then:

g'(a) = lim ( 1/(Δx) - 1/0 ) / Δx therefore not only is 1/0 not defined but 1/Δx does not have a limit near 0 in

Δx→0

this case so the limit as Δx → 0 does not exist.

Am I on to something or am I way off base?

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# Continuity question

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