Why is $f(x)=\frac{\sin x}{x^{-1}}$ defined when $x=0?$ 
Why is $f(x)=\frac{\sin x}{x^{-1}}$ defined when $x=0?$

There's something that confuses me:
If we take the function $f(x)=\frac{\sin x}{x^{-1}}$ then we can say it is equal to $f(x)=x\sin x$ and therefore $f(0)=0$. But if we try to get $f(0)$ from the original function we get $f(0)=\dfrac{\sin 0}{\frac{1}{0}}$ which I'd expect to be undefined just like $\frac{\sin 0}{0}$ is undefined. So how come it is defined and equal to $0?$
 A: There is no hesitation, 
$$\frac{\sin(x)}{x^{-1}}$$ is not defined at $x=0$ because the denominator is not defined.

You may not claim that
$$\frac{\sin(x)}{x^{-1}}=x\sin(x)$$ at $x=0$, though
$$\lim_{x\to0}\frac{\sin(x)}{x^{-1}}=\lim_{x\to0}x\sin(x)$$ is true.

For the same reason,
$$x\sin(x)=\begin{cases}0,&x=0,\\\dfrac{\sin(x)}{x^{-1}},&x\ne0\end{cases}$$ is true.
A: Well, one could "simply" say that it is:
$$x=\frac{1}{x^{-1}} = \frac{1}{\frac{1}{x}} $$
One could claim that the function $f(x) = x$ is defined for all $x \in \mathbb R$ but in order to define at $x=0$ it is a must that one should be more rigorous. Keep in mind, though, that expressing things as fractions results in restrictions for the given variables. Thus, in order to revert to an expression such as the equalities at the RHS of the first line, one should apply specific restrictions.
It is just a matter of convention to be honest. I prefer to be rigorous and 100% correct. Thus I would not say that $\dfrac{1}{x^{-1}}$ is defined at $x=0$.
If you want to express the function $f(x) = x$ as the fraction above, then you may exclude the case of $x=0$ and note that its value is:
$$f(0) = \lim_{x \to 0^-} \frac{1}{x^{-1}} = \lim_{x \to 0^+} \frac{1}{x^{-1}}.$$
The limits are easy to calculate.
Then the function
$$f(x) = \begin{cases} 1 &, \ x =0 \\ \dfrac{1}{x^{-1}} &,x \neq 0 \end{cases}$$
is defined rigorously.
Edit: As I saw your comment about online calculators, it is important to mention that computative systems tend to simplify expressions sometimes, not taking into account such sensitive matters. Thus, we should always revert to rigorous terms to be sure and precise.
