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Let's analyze the options one by one to determine which function has a limit of 0 as \( x \to \infty \):

1. \( f(x) = x^2 \): As \( x \to \infty \), \( x^2 \) grows without bound, so the limit is \( \infty \), not 0.
2. \( f(x) = \cos(x) \): The function \( \cos(x) \) oscillates between -1 and 1 as \( x \to \infty \), so it does not settle to 0.
3. \( f(x) = 2^{-x} \): As \( x \to \infty \), \( 2^{-x} = \frac{1}{2^x} \) approaches 0 because the denominator grows exponentially.
4. \( f(x) = x \): As \( x \to \infty \), \( x \) grows without bound, so the limit is \( \infty \), not 0.
5. \( f(x) = \ln(x) \): As \( x \to \infty \), \( \ln(x) \) also grows without bound, although more slowly than \( x \).

The correct answer is:  
**\( f(x) = 2^{-x} \)** since this function approaches 0 as \( x \to \infty \).

Would you like a more detailed explanation or have any further questions?

Here are five related questions to further your understanding:
1. What is the general behavior of exponential functions like \( a^{-x} \) as \( x \to \infty \)?
2. How do logarithmic functions behave compared to linear functions as \( x \to \infty \)?
3. Can a trigonometric function like \( \cos(x) \) ever have a limit as \( x \to \infty \)?
4. What is the difference between functions that grow polynomially and those that grow exponentially?
5. Why does \( x^2 \) grow faster than \( \ln(x) \) as \( x \to \infty \)?

**Tip:** When analyzing limits, it's helpful to recognize the general growth rates of common functions like polynomials, exponentials, and logarithms.

Which of the following functions has a limit of 0 as x goes to infinity?

Explore which functions have a limit of 0 as x approaches infinity, covering exponential, logarithmic, and trigonometric functions. This problem demonstrates exponential decay, polynomial growth, and limit behavior, suitable for Grades 10-12.

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