To graph the behavior where $\lim_{x \to -\infty} f(x) = \infty$, we visualize a function where the values increase without bound as $x$ approaches $-\infty$.

Steps:

1. Choose a function that demonstrates this behavior, such as $f(x) = -x$.
2. Graph $f(x) = -x$. As $x$ becomes more negative (approaching $-\infty$), $f(x)$ becomes larger and larger (approaching $+\infty$).

Let me plot this for you.import numpy as np import matplotlib.pyplot as plt

Define the function

x = np.linspace(-50, 50, 500) y = -x

Plot the graph

plt.figure(figsize=(8, 6)) plt.plot(x, y, label="f(x) = -x", linewidth=2) plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.title("Graph of f(x) = -x (lim x→-∞ f(x) = ∞)") plt.xlabel("x") plt.ylabel("f(x)") plt.grid(alpha=0.5) plt.legend() plt.show() <<ImageDisplayed>>successHere is the graph of $f(x) = -x$, which illustrates that as $x \to -\infty$, $f(x) \to \infty$. The function's slope ensures that its values increase without bound as $x$ approaches negative infinity.

Would you like further analysis, or should I clarify something specific?

Here are 5 related questions to explore:

1. What are some other functions where $\lim_{x \to -\infty} f(x) = \infty$?
2. How do we formally prove limits involving $\pm \infty$?
3. Can this concept be applied to real-world problems? If so, how?
4. What happens if we switch to $\lim_{x \to +\infty} f(x) = \infty$?
5. What are some functions where $\lim_{x \to -\infty} f(x) = -\infty$?

Tip: When analyzing limits, visualizing them with graphs can often make their behavior easier to understand!