1. **State the problem:** We are given a function $R(x)$ with a graph showing vertical asymptotes at $x = -3$, $x = 2$, and $x = 5$. We need to find the limits of $R(x)$ as $x$ approaches these points and identify the equations of the vertical asymptotes. 2. **Recall the definition of limits near vertical asymptotes:** - If the function approaches infinity or negative infinity as $x$ approaches a value, the limit is infinite or does not exist (DNE) in the finite sense. - Vertical asymptotes occur where the function grows without bound. 3. **Evaluate each limit:** (a) $\lim_{x \to 2} R(x)$ - From the graph, as $x \to 2^-$, $R(x) \to -\infty$. - As $x \to 2^+$, $R(x) \to +\infty$. - Since the left and right limits differ, the two-sided limit does not exist. (b) $\lim_{x \to 5} R(x)$ - As $x \to 5^-$, $R(x) \to +\infty$. - As $x \to 5^+$, $R(x) \to -\infty$. - Left and right limits differ, so the limit does not exist. (c) $\lim_{x \to -3} R(x)$ - As $x \to -3^-$, $R(x) \to -\infty$. - As $x \to -3^+$, $R(x) \to +\infty$. - Left and right limits differ, so the limit does not exist. (d) $\lim_{x \to -3^+} R(x)$ - From above, $\lim_{x \to -3^+} R(x) = +\infty$. 4. **Vertical asymptotes:** - The graph shows vertical asymptotes at $x = -3$, $x = 2$, and $x = 5$. - These are the smallest, middle, and largest values respectively. **Final answers:** (a) $\lim_{x \to 2} R(x) = \text{DNE}$ (b) $\lim_{x \to 5} R(x) = \text{DNE}$ (c) $\lim_{x \to -3} R(x) = \text{DNE}$ (d) $\lim_{x \to -3^+} R(x) = +\infty$ (e) Vertical asymptotes: $x = -3$, $x = 2$, $x = 5$