Is sinx/x a transcendental function?

Question

Or is it algebraic? Is there any properties of transcendental functions that allow me to recognize them?

Answer 1

If $f(x)$ is an algebraic function, then so is $x\cdot f(x)$. However, if we mutliply this function by $x$, we obtain $\sin x$, which is one of the basic examples of a transcendental function. Therefore also, $\frac{\sin x}{x}$ is transcendental.

Answer 2

If $\displaystyle \frac {\sin x} x$ is algebraic, then so would be $\sin x$, since $x$ is algebraic and product of algebraic functions are algebraic.