Maximal solution of $x'(t) = t^{-2}\cos(t^{-1})$

Question

Considering the cauchy equation $x'(t)=t^{-2}\cos(t^{-1}),x\left(\frac{2}{\pi}\right)=1$, I've found that $x(t)=\sin(t^{-1})$. The second problem is to find the domain of maximal solution, I think that the domain is $(0,\infty)$ and proved that the right limit is $\infty$, but I don't know how to prove that the left limit is $0$, because there is no $\lim_{t \rightarrow 0^+}\sin(t^{-1})$.

Answer 1

The solution is defines for $t>0$. The maximal iterval of existence is $(0,+\infty)$. The right limit of that interval is $+\infty$, and the left limit is $0$. The use of the word limit in here does not refer to the limit of the solution as $t\to0^+$, but to the limit (or extreme, or boundary,...) of the maximal interval of existence.