109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
e. f(x) = arccsc(x), g(x) = 1/x

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Recall the definitions and behavior of the functions as \( x \to \infty \): \( f(x) = \arccsc(x) \) and \( g(x) = \frac{1}{x} \).

Understand that \( \arccsc(x) = \arcsin\left(\frac{1}{x}\right) \). As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( f(x) = \arcsin\left(\frac{1}{x}\right) \) approaches \( \arcsin(0) = 0 \).

Use the approximation for small angles: \( \arcsin(y) \approx y \) when \( y \to 0 \). Therefore, \( f(x) \approx \frac{1}{x} \) for large \( x \).

Since \( g(x) = \frac{1}{x} \), both \( f(x) \) and \( g(x) \) behave like \( \frac{1}{x} \) as \( x \to \infty \).

Conclude that \( f(x) \) and \( g(x) \) grow at the same rate as \( x \to \infty \) because their leading behavior is proportional to \( \frac{1}{x} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Asymptotic Behavior of Functions

Asymptotic behavior describes how functions behave as the input grows very large (x→∞). Understanding this helps compare growth rates by analyzing limits or dominant terms, revealing which function increases or decreases faster or if they behave similarly.

Inverse Trigonometric Functions and Their Limits

The arccsc(x) function is the inverse of the cosecant function, defined for |x| ≥ 1. As x→∞, arccsc(x) approaches zero because csc(θ) grows large near zero angles. Knowing this limit helps compare arccsc(x) to other functions like 1/x.

Limit Comparison for Growth Rates

To compare growth rates of f(x) and g(x), evaluate the limit of their ratio as x→∞. If the limit is zero, infinity, or a finite nonzero number, it indicates whether f grows slower, faster, or at the same rate as g, respectively.

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