I'm a grad student in Canada and I'm having some trouble with these supplemental problems. I've never really had a grasp on the difference between cluster points and limits, so these questions are stumping me:
Let there be some set W in $\mathbb{R}$, and there are two functions $f$ and $h$. There is a cluster point $c$ of $W$. $\lim_{x\to c} f $ exists, while $\lim_{x\to c} g $ does not exist.
I'm wondering whether $\lim_{x\to c} f-g $ exists or not, and whether $\lim_{x\to c} f*g $ exists or not, and also why.
Thank you so much for the help!
Note that a cluster point of a set $S$ (also known as a limit point) is just a point $x$ s.t. for any open set $U$ containing $x$, $(U \cap S)-\{x\} \neq \emptyset$. The reason we want to take $c$ a cluster point is those are the points for which it makes sense to take limits: precisely the cluster points of the domain (you should try to convince yourself of that).
I will now give examples of showing that sometimes the $f \cdot g$ limit exists and sometimes it doesn't.
Now let $W = \mathbb{R}$ and $c=0$. Consider $f(x) := 0$ and $g(x) = \frac{1}{x}\sin(x)$. Then $\lim_{x \to 0} f(x)$ exists and is equal to 0. The limit for $g$ does not exist. Then $\lim_{x\to 0} f \cdot g(x) = 0$ exists. Take $h(x) =1$. Then $\lim_{x\to 0} h(x) = 1$ exists. But $\lim_{x \to 0} h \cdot g(x) = \lim_{x \to 0} g(x)$ does not exist.
Suppose that the limit $\lim_{x \to c} f(x)$ and $\lim_{x \to c} f-g(x)$ both exist, then $\lim_{x \to c} f(x) - \lim_{x \to c} f- g(x) = \lim_{x \to c} g(x)$ exists, by additivity of limits. Thus if the $g$ limit doesn't exists but the $f$ limit exists, then $\lim_{x \to c} f-g$ does not exist.