How can the function f(x) = x/sin(x) for x != 0 be redefined at x=0 to make it continuous? What is the value?

For a function $f(x)$ to be continuous at a point $x=c$, the limit of the function as $x$ approaches $c$ must be equal to the function's value at $c$. That is, $\lim_{x \to c} f(x) = f(c)$.
In this problem, the function is $f(x) = \frac{x}{\sin x}$ for $x \neq 0$, and we need to redefine it at $x=0$ to make it continuous. So, we need to find the value of $\lim_{x \to 0} f(x)$.

The function is $f(x) = \frac{x}{\sin x}$. We need to find:

This is a standard trigonometric limit. We know the fundamental limit:

We can rewrite the given limit using this identity:

As $x \to 0$, the denominator $\frac{\sin x}{x}$ approaches 1. Therefore:

For the function $f(x)$ to be continuous at $x=0$, we must define $f(0)$ to be equal to the limit we just calculated.
Therefore, to make the function continuous at $x=0$, we must define $f(0) = 1$.

The value that the function can be redefined to at $x=0$ to make it continuous is $\boxed{1}$.