1. **Problem Statement:** We are analyzing different types of discontinuities in functions at specific points: at $x=-1$, $x=1$, $x=2$, and $x=0$. 2. **Key Concepts:** - A **vertical asymptote** occurs where the function grows without bound (to $+\infty$ or $-\infty$). - A **jump discontinuity** occurs when the left-hand and right-hand limits exist but are not equal. - A **hole** (removable discontinuity) occurs when the limit exists but the function is not defined or differs at that point. - **Continuity** means the function value equals the limit at that point. 3. **At $x=-1$ (Vertical Asymptote and Hole):** - The function shoots to $+\infty$ as $x \to -1^-$. - The function approaches $0$ as $x \to -1^+$. - There is a hole at $(-1,0)$. 4. **At $x=1$ (Hole):** - The limit as $x \to 1$ is $-1$. - The function is not defined at $x=1$ (hole). 5. **At $x=2$ (Jump Discontinuity):** - Left limit: $\lim_{x \to 2^-} f(x) = -2$. - Right limit: $\lim_{x \to 2^+} f(x) = 2$. - The function jumps from $-2$ to $2$ at $x=2$. 6. **At $x=0$ (Continuous):** - The function value is $f(0) = 1$. - The limit as $x \to 0$ is $1$. - The function is continuous at $x=0$. 7. **Summary:** - Vertical asymptote at $x=-1$ with a hole at $(-1,0)$. - Hole at $x=1$ with limit $-1$ but no function value. - Jump discontinuity at $x=2$ from $-2$ to $2$. - Continuous at $x=0$ with value $1$. This analysis clarifies the behavior of the function at these points and the types of discontinuities present.