Subjects calculus

Limit Vertical Asymptotes B26D59

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1. **State the problem:** We need to determine which of the given limit expressions agree with the behavior of the graph near the vertical asymptotes at $x = 1$, $x = 3$, and $x = 5$. 2. **Recall the meaning of limits approaching infinity:** - $\lim_{x \to a^-} f(x) = -\infty$ means as $x$ approaches $a$ from the left, $f(x)$ decreases without bound. - $\lim_{x \to a^+} f(x) = -\infty$ means as $x$ approaches $a$ from the right, $f(x)$ decreases without bound. 3. **Analyze each limit:** - (A) $\lim_{x \to 1^-} f(x) = -\infty$: The graph shows the function descending steeply to negative infinity as $x$ approaches 1 from the left. So, (A) is **true**. - (B) $\lim_{x \to 3^+} f(x) = -\infty$: The graph shows the function descending steeply to negative infinity as $x$ approaches 3 from the right. So, (B) is **true**. - (C) $\lim_{x \to 5^-} f(x) = -\infty$: The graph shows the function descending steeply to negative infinity as $x$ approaches 5 from the left. So, (C) is **true**. **Final answer:** All three limits (A), (B), and (C) agree with the graph. $$\boxed{\text{A, B, and C are all correct}}$$