Existence of a limit and a counterexample For homework, my group was given this false statement which we were supposed to find an example for:
If a function $g$ is bounded from top and bottom on an open interval $(a,b)$, then there exists a real number (non infinite) limit:
$$\lim_{x\to b^-} f(x) $$
We were trying to come up with a solution but we think that there is no counterexample, as when a function is bounded from both sides, it must have a real number lmit at the ends of the function. Are we wrong?
(As requested by the OP)
It seems to me that $f(x) = \sin(1/x)$ considered on the $(-1,0)$ interval provides a counter-example, since $f(x)$ is bounded on the interval, and $\lim_{x \rightarrow 0^-} f(x)$ is undefined.
The known property is that a limit exists for a bounded and monotonic function. Your function can fail to be monotonic in any neighborhood of $b$, as shown by @user3733558.