I had an idea using
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Can we take the limit as x-> c of ?
Let g: A --> R and assume that f is a bounded function on A, which is a subset of R (i.e. there exists M > 0 satisfying |f(x)| <= M for all x in A). Show that if lim as x-->c of g(x) = 0, then lim as x-->c of g(x)*f(x) = 0 as well.