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If lim x→c =L, then f(c) = L.

I don't think that is true because f(c) may not always equal L? or is the statement true? I'm not sure

Also:

If f(x) < g(x) for all x≠a, then lim x→a f(x) < lim x→a g(x)

I'm not sure about this one either. I know that if f(x)=g(x) for all x≠c in an open interval containing c, and their limits exists, then lim x→c f(x) = lim x→c g(x). But I don't know if it's is the same for this inequality.