Applying limit inferior

Question

In the solution to this question while attempting to prove by contradiction it states that if we assume $|x(t)|\nrightarrow \infty $ then $M:=$lim inf $_{t\uparrow b}|x(t)|$

Why can I just not use the fact that $|x(t)|\leq M$ as $t \uparrow b$

Answer 1

Because a function may fail to converge to $+\infty$ at $b$ AND be unbounded in every neighbourhood of $b$.

For an example, assume that $b=0$ and consider the function defined, for every $t\ne0$, by $$x(t)=\frac2t\,\sin\left(\frac\pi t\right).$$ Then, for every $n\geqslant0$, $$x\left(\frac1{n+1}\right)=0,\qquad x\left(\frac2{4n+1}\right)=4n+1,$$ hence the function $|x|$ neither converges to $+\infty$ at $0$ nor is bounded in a neighbourhood of $0$.