If f is defined as a function of real variables to real values, and c∈cl(Domain) as its limit value (i.e. limx→cf(x)=0) how to prove that this implies: limx→c1f(x)=∞.
It seems logical that the values will be always bigger, but when tried to construct a contradiction using the y-creterion I stuck at: ∃ϵ>0:f(x)>0∀x∈[c−ϵ,c+ϵ].
Answer
This is problematic, even if you consider 1/|f(x)| instead of 1/f(x). For example, let f(x)={xsin(1/x)x≠00otherwise.
We must make some extra assumptions to take care of your problems. In particular, you need to show the following:
Suppose that E⊆R and f:E→R. Let F={x∈E:f(x)≠0}. Suppose further that c∈R is a limit point of both E and F, and that for all ϵ>0 there is some δ>0 such that |f(x)|<ϵ whenever x∈E with 0<|x−c|<δ. Then for all M, there exists δ>0 such that 1/|f(x)|>M whenever x∈F with 0<|x−c|<δ.
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