Sketch a possible graph for a function 𝑓 satisfying all specified properties below. If
there is an asymptote, you must graph the asymptote. Label all essential information.
a. The domain of 𝑓 is (−∞, ∞)
b. lim 𝑥 → −∞
𝑓(𝑥) = ∞
c. lim 𝑥 → −4
𝑓(𝑥) = 5
d. 𝑓(−4) = −2
e. 𝑓′(−2) > 0
f. 𝑓 has an infinite discontinuity at 1
g. 𝑓 is left-continuous at 3, but not continuous at 3
h. lim𝑥 → ∞
𝑓(𝑥) = 5

The graph provided represents a function \( f(x) \) with the following characteristics:

- The domain of \( f \) is \( (-\infty, \infty) \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
- As \( x \to -4 \), \( f(x) \to 5 \), but \( f(-4) = -2 \).
- The derivative \( f'(-2) > 0 \), meaning the function is increasing at \( x = -2 \).
- There is an infinite discontinuity at \( x = 1 \), indicating a vertical asymptote.
- The function is left-continuous but not continuous at \( x = 3 \).
- As \( x \to \infty \), \( f(x) \to 5 \), suggesting a horizontal asymptote at \( y = 5 \).

Let me know if you need further clarification or details about any specific part of the graph.

Here are 5 related questions:
1. What is the significance of left-continuity at \( x = 3 \)?
2. How would you describe the behavior of the function around \( x = 1 \)?
3. How does the function's increasing nature around \( x = -2 \) affect its graph?
4. What does the horizontal asymptote at \( y = 5 \) tell you about the long-term behavior of the function?
5. How does an infinite discontinuity impact the overall behavior of the function?

**Tip:** Infinite discontinuities often indicate vertical asymptotes, where the function grows without bound as it approaches a particular value of \( x \).

Sketch a possible graph for a function f satisfying all specified properties below. If there is an asymptote, you must graph the asymptote. Label all essential information.
a. The domain of f is (-∞, ∞)
b. lim x → -∞ f(x) = ∞
c. lim x → -4 f(x) = 5
d. f(-4) = -2
e. f′(-2) > 0
f. f has an infinite discontinuity at 1
g. f is left-continuous at 3, but not continuous at 3
h. lim x → ∞ f(x) = 5

Sketch the graph of a function that satisfies conditions including limits at infinity, infinite discontinuities, left-continuity, and a derivative indicating an increasing interval. The problem involves understanding key calculus concepts such as asymptotes, discontinuities, and derivative properties.

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