1. **State the problem:** We want to investigate the limit $$\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right).$$ 2. **Recall the function and its behavior:** The function is $$f(x) = \sin\left(\frac{\pi}{x}\right),$$ which is undefined at $x=0$ because division by zero is undefined. 3. **Evaluate the function at some points approaching 0:** - $f(1) = \sin(\pi) = 0$ - $f\left(\frac{1}{2}\right) = \sin(2\pi) = 0$ - $f\left(\frac{1}{3}\right) = \sin(3\pi) = 0$ - $f\left(\frac{1}{4}\right) = \sin(4\pi) = 0$ - $f(0.1) = \sin(10\pi) = 0$ - $f(0.01) = \sin(100\pi) = 0$ 4. **Initial guess:** Since $f\left(\frac{1}{n}\right) = \sin(n\pi) = 0$ for any integer $n$, one might guess the limit is 0. 5. **However, the function oscillates:** There are infinitely many values of $x$ approaching 0 where $f(x) = 1$ or $f(x) = -1$, for example when $$\frac{\pi}{x} = \frac{\pi}{2} + 2k\pi \implies x = \frac{2}{4k+1}$$ for integers $k$, making $\sin\left(\frac{\pi}{x}\right) = 1$. 6. **Conclusion:** Because the function oscillates between $-1$ and $1$ infinitely often as $x \to 0$, the limit does not approach a single value. **Final answer:** $$\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right) \text{ does not exist.}$$