$$\lim_{z \to \infty}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = 1$$
$$\lim_{z \to 0^-}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = 0$$
More at z→0 from the left$$\lim_{z \to 0^+}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = 0$$
More at z→0 from the right$$\lim_{z \to 1^-}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = \sin{\left(1 \right)}$$
More at z→1 from the left$$\lim_{z \to 1^+}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = \sin{\left(1 \right)}$$
More at z→1 from the right$$\lim_{z \to -\infty}\left(z \sin{\left(1 \cdot \frac{1}{z} \right)}\right) = 1$$
More at z→-oo