1. **Problem Statement:** Prove that $$\lim_{t \to 0} \sin t = 0$$. 2. **Recall the definition of limit:** For a function $f(t)$, $$\lim_{t \to a} f(t) = L$$ means that as $t$ approaches $a$, $f(t)$ approaches $L$. 3. **Key fact about sine:** The sine function is continuous everywhere, and specifically at $t=0$, $\sin 0 = 0$. 4. **Using the Squeeze Theorem:** We know for all real $t$, $$-1 \leq \sin t \leq 1$$. 5. **Near zero, sine behaves like its argument:** Using the inequality $$|\sin t| \leq |t|$$ for all $t$. 6. **Apply limit properties:** Since $$\lim_{t \to 0} |t| = 0$$, by the Squeeze Theorem, $$- |t| \leq \sin t \leq |t|$$ and both $$\lim_{t \to 0} -|t| = 0$$ and $$\lim_{t \to 0} |t| = 0$$. 7. **Conclusion:** Therefore, $$\lim_{t \to 0} \sin t = 0$$. This completes the proof that the limit of sine as $t$ approaches zero is zero.