1. **Stating the problem:** We are given a function $y=f(x)$ with vertical asymptotes at $x=-2$, $x=0$, and $x=2$. We need to find the following limits: (a) $\lim_{x \to -2} f(x)$ (b) $\lim_{x \to 0^-} f(x)$ (c) $\lim_{x \to 0^+} f(x)$ (d) $\lim_{x \to 2^-} f(x)$ (e) $\lim_{x \to 2^+} f(x)$ (f) The vertical asymptotes of $f$. 2. **Concepts and theory:** - A vertical asymptote at $x=a$ means $f(x)$ approaches $\pm \infty$ as $x$ approaches $a$ from the left or right. - The limit $\lim_{x \to a} f(x)$ may not exist if the left and right limits differ. - If there is a filled point at $(a,b)$, then $f(a)=b$. - If there is an open circle at $(a,b)$, then $f(a) \neq b$ and the function is not defined at that point. 3. **Step-by-step solution:** (a) $\lim_{x \to -2} f(x)$: - Since there is a vertical asymptote at $x=-2$, the limit tends to infinity or negative infinity. - From the graph, as $x \to -2$ from the left, $f(x) \to -\infty$ and from the right, $f(x) \to +\infty$. - Because the left and right limits differ, the limit does not exist. (b) $\lim_{x \to 0^-} f(x)$: - There is a vertical asymptote at $x=0$. - From the left side, $f(x)$ approaches $+\infty$. (c) $\lim_{x \to 0^+} f(x)$: - From the right side of $x=0$, $f(x)$ approaches $-\infty$. (d) $\lim_{x \to 2^-} f(x)$: - There is a vertical asymptote at $x=2$. - From the left side, $f(x)$ approaches $+\infty$. (e) $\lim_{x \to 2^+} f(x)$: - From the right side, $f(x)$ approaches $-\infty$. (f) **Vertical asymptotes:** - The vertical asymptotes are at $x=-2$, $x=0$, and $x=2$ as given. 4. **Summary of answers:** - (a) $\lim_{x \to -2} f(x)$ does not exist because left and right limits differ. - (b) $\lim_{x \to 0^-} f(x) = +\infty$ - (c) $\lim_{x \to 0^+} f(x) = -\infty$ - (d) $\lim_{x \to 2^-} f(x) = +\infty$ - (e) $\lim_{x \to 2^+} f(x) = -\infty$ - (f) Vertical asymptotes at $x=-2$, $x=0$, and $x=2$. This completes the detailed explanation and solution for the exercise.