Let I R be an open interval let c I and let f g I c R be f

Let I R be an open interval, let c I and let f, g : I – {c} R be functions. Suppose that lim(xc)f(x) exists and that lim(xc)g(x) does not exist.

(1) Prove that lim(xc)[f+g](x) does not exist.

(2) Prove that if lim(xc)f(x) 0, then lim(xc)[fg](x) does not exist.

All limits are taken as x->c in what follows.

(1) Assume lim f(x) exists and lim g(x) does not exist.

To show lim(f(x)+g(x)) does not exist.

Assume it exists. then

lim g(x) = lim (f(x)+g(x))-lim f(x) (as both exist)

This implies lim g(x) exists contrary to the assumption of g(x)

Hence lim(xc)[f+g](x) does not exist.

(ii) If both lim(xc)[fg](x) and lim(xc)f(x) exist, then

lim(xc) g(x) = lim(xc) [fg(x)]/lim(xc)f(x)) (well defined as the denominator is not 0, by given)

would exist , contrary to the assumption on g(x)

Hence, lim(xc)[fg](x) does not exist.