Function Limits B6D179

1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches infinity, negative infinity, and the points $x=1$ and $x=3$. We also need to find the equations of the asymptotes based on the graph description. 2. **Recall limit and asymptote concepts:** - The limit $\lim_{x \to \infty} f(x)$ describes the behavior of $f(x)$ as $x$ becomes very large. - The limit $\lim_{x \to -\infty} f(x)$ describes the behavior as $x$ becomes very negative. - Vertical asymptotes occur where the function tends to $\pm \infty$ as $x$ approaches a finite value. - Horizontal asymptotes describe the value the function approaches as $x \to \pm \infty$. 3. **Analyze each limit:** (a) $\lim_{x \to \infty} f(x)$: The graph slowly decreases to the right after a small hump, so the function approaches a horizontal asymptote. The limit is a finite number, say $L$. (b) $\lim_{x \to -\infty} f(x)$: The graph approaches a horizontal asymptote as $x \to -\infty$. The limit is a finite number, say $M$. (c) $\lim_{x \to 1} f(x)$: The graph has a vertical asymptote at $x=1$ and rises sharply near $x=1$. This means the limit tends to $\pm \infty$. We check the left and right limits: - From the left, $f(x) \to \infty$. - From the right, $f(x) \to -\infty$. (d) $\lim_{x \to 3} f(x)$: The graph has a vertical asymptote at $x=3$ and drops to negative infinity near $x=3$. So: - From the left, $f(x) \to -\infty$. - From the right, $f(x) \to \infty$. 4. **Equations of the asymptotes:** - Vertical asymptotes at $x=1$ and $x=3$. - Horizontal asymptote is the value the function approaches as $x \to \pm \infty$. From the graph description, the function slowly decreases to the right, so the horizontal asymptote is $y=0$. 5. **Summary of answers:** \begin{align*} &\text{(a) } \lim_{x \to \infty} f(x) = 0 \\ &\text{(b) } \lim_{x \to -\infty} f(x) = 0 \\ &\text{(c) } \lim_{x \to 1^-} f(x) = \infty, \quad \lim_{x \to 1^+} f(x) = -\infty \\ &\text{(d) } \lim_{x \to 3^-} f(x) = -\infty, \quad \lim_{x \to 3^+} f(x) = \infty \\ &\text{(e) Vertical asymptotes: } x=1, x=3; \quad \text{Horizontal asymptote: } y=0 \end{align*}