1. **State the problem:** Find the asymptotes of the function $y=\frac{\sin x}{x}$.\n\n2. **Recall the definition of asymptotes:** Asymptotes are lines that the graph of a function approaches but never touches as $x$ approaches a certain value or infinity. There are three types: vertical, horizontal, and oblique.\n\n3. **Check for vertical asymptotes:** Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. Here, the denominator is zero at $x=0$. However, $\sin 0=0$, so the function is of the form $\frac{0}{0}$ at $x=0$. Using the limit:\n$$\lim_{x \to 0} \frac{\sin x}{x} = 1,$$\nwhich is finite, so there is no vertical asymptote at $x=0$. Instead, the function has a removable discontinuity there.\n\n4. **Check for horizontal asymptotes:** Horizontal asymptotes are found by evaluating limits as $x \to \pm \infty$.\n$$\lim_{x \to \infty} \frac{\sin x}{x} = 0,$$\nbecause $\sin x$ oscillates between $-1$ and $1$, and dividing by an infinitely large $x$ makes the fraction approach zero. Similarly,\n$$\lim_{x \to -\infty} \frac{\sin x}{x} = 0.$$\nThus, the line $y=0$ is a horizontal asymptote.\n\n5. **Check for oblique asymptotes:** Since the degree of the numerator is less than the denominator, no oblique asymptotes exist.\n\n**Final answer:** The function $y=\frac{\sin x}{x}$ has a horizontal asymptote at $y=0$ and no vertical or oblique asymptotes.