The question asks to determine which statements must be true based on the given information that:

$\lim_{x \to \infty} f(x) = 2$

Let's analyze each option:

A. $\lim_{x \to -\infty} f(x) = -2$

This is unrelated to the provided information because the given limit only specifies what happens as $x \to \infty$ (positive infinity), not as $x \to -\infty$ (negative infinity). Therefore, A is not necessarily true.

B. $\lim_{x \to -\infty} f(x) = 2$

Again, this concerns the behavior of $f(x)$ as $x \to -\infty$, but the problem only tells us about the limit as $x \to \infty$. There’s no guarantee that the function behaves the same way in the negative direction. Thus, B is not necessarily true.

C. $f(x)$ is continuous at $x = \infty$

Continuity at $x = \infty$ is not a meaningful concept because continuity applies to finite values of $x$, and we don't assess continuity at infinity. Therefore, C is false.

D. $f(x)$ has a horizontal asymptote at $y = 2$

The limit $\lim_{x \to \infty} f(x) = 2$ means that as $x$ gets very large, $f(x)$ approaches 2. This is the definition of a horizontal asymptote at $y = 2$. Hence, D is true.

E. As $x$ gets very large, $f(x)$ gets close to 2

This directly follows from the limit definition $\lim_{x \to \infty} f(x) = 2$. As $x$ becomes very large, $f(x)$ approaches 2. Thus, E is true.

F. $f(x)$ has a vertical asymptote at $x = 2$

A vertical asymptote means that the function becomes unbounded as $x$ approaches a certain value, but this statement doesn’t follow from the given information. The limit tells us about the behavior as $x \to \infty$, not about any vertical asymptotes. Therefore, F is false.

G. $f(x)$ is never equal to 2

The limit $\lim_{x \to \infty} f(x) = 2$ only means that $f(x)$ approaches 2 as $x$ becomes large, but it doesn’t imply that $f(x)$ can never actually equal 2. Thus, G is false.

Conclusion:

The correct answers are:

• D: $f(x)$ has a horizontal asymptote at $y = 2$.
• E: As $x$ gets very large, $f(x)$ gets close to 2.

Would you like more details on any part of the explanation?

Related Questions:

1. What is the formal definition of a horizontal asymptote?
2. Can a function have different limits as $x \to \infty$ and $x \to -\infty$?
3. How does continuity relate to limits?
4. What is the difference between horizontal and vertical asymptotes?
5. How can we verify if a function actually equals its limit value at infinity?

Tip: Horizontal asymptotes describe the end behavior of a function as $x$ approaches infinity or negative infinity, indicating that the function gets arbitrarily close to a particular value but doesn't necessarily reach it.