Math 351 (Real Analysis) Solutions to the Sixth Homework Set. 1 ...

4. Give δ − ɛ definitions for the one-sided limits lim f(x) = L and lim f(x) = M.
x→c− x→c +
Solution: We say that lim f(x) = L if given any ɛ > 0, there exists δ > 0 such that
x→c− whenever c − δ < x, |f(x) − L| < ɛ.
Likewise, lim f(x) = M if given any ɛ > 0, there exists δ > 0 such that whenever
x→c +
x < c + δ, |f(x) − M| < ɛ.
5. Let g : A → R and assume that f is a bounded function on A ⊆ R (i.e., there exists
M > 0 satisfying |f(x)| ≤ M for all x ∈ A). Show that if lim g(x) = 0, then
x→c
lim g(x)f(x) = 0 as well.
x→c
Solution: By hypothesis on f, there exists M > 0 satisfying |f(x)| ≤ M for all
x ∈ A. Moreover, given ɛ > 0, there exists δ > 0 such that |x − c| < δ implies
|g(x) − 0| < ɛ/M.
Thus, |x − c| < δ implies that |g(x)f(x) − 0| < (ɛ/M)M = ɛ, as required.