1. **Problem Statement:** We need to sketch the function $f(x) = \frac{1}{x}$ and identify its discontinuities. 2. **Function and Important Rules:** The function is $f(x) = \frac{1}{x}$. This is a rational function where the denominator cannot be zero because division by zero is undefined. 3. **Identify Discontinuities:** The denominator $x$ is zero at $x=0$, so the function is undefined there. This means there is a discontinuity at $x=0$. 4. **Behavior Near the Discontinuity:** - As $x \to 0^+$, $f(x) = \frac{1}{x} \to +\infty$. - As $x \to 0^-$, $f(x) = \frac{1}{x} \to -\infty$. This shows a vertical asymptote at $x=0$. 5. **Behavior at Infinity:** - As $x \to +\infty$, $f(x) = \frac{1}{x} \to 0^+$. - As $x \to -\infty$, $f(x) = \frac{1}{x} \to 0^-$. This shows a horizontal asymptote at $y=0$. 6. **Sketching the Graph:** - The graph has two branches, one in the first quadrant approaching $y=0$ as $x \to +\infty$ and going to $+\infty$ as $x \to 0^+$. - The other branch is in the third quadrant approaching $y=0$ as $x \to -\infty$ and going to $-\infty$ as $x \to 0^-$. **Final answer:** The function $f(x) = \frac{1}{x}$ has a discontinuity at $x=0$ (vertical asymptote). It also has a horizontal asymptote at $y=0$.