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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