$$\frac{\sin \left(x\right)}{x}$$
Hello, I know how to prove without the definition, I tried a lot doing it by the definition of uniform continuity $\left|\frac{\sin \:\left(x\right)}{x}-\frac{\sin \:\left(y\right)}{y}\right|<\xi $ for every $x,y$ close enough.
Thank you very much.