Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
19. lim(x→∞)arccsc(x)

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Recall the definition of the function involved: \(\arccsc(x)\) is the inverse cosecant function, which gives the angle \(\theta\) such that \(\csc(\theta) = x\).

Understand the domain and range of \(\arccsc(x)\): the domain is \((-\infty, -1] \cup [1, \infty)\), and the range is \([-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}]\) excluding zero.

Analyze the behavior of \(\arccsc(x)\) as \(x\) approaches infinity: since \(\csc(\theta) = \frac{1}{\sin(\theta)}\), large values of \(x\) correspond to \(\sin(\theta)\) approaching zero from the positive side.

Determine the angle \(\theta\) whose cosecant is very large positive: this angle approaches \(0\) from the positive side, because \(\csc(\theta)\) becomes large when \(\sin(\theta)\) is close to zero but positive.

Conclude that \(\lim_{x \to \infty} \arccsc(x)\) approaches \(0\) from the positive side, so the limit is \(0^+\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Function as x Approaches Infinity

This concept involves finding the value that a function approaches as the input variable x grows without bound. Understanding limits at infinity helps determine the end behavior of functions, which is crucial for analyzing asymptotic behavior and horizontal asymptotes.

Inverse Trigonometric Functions - Arccsc(x)

Arccsc(x) is the inverse cosecant function, which returns the angle whose cosecant is x. Its domain is |x| ≥ 1, and its range is typically restricted to [-π/2, 0) ∪ (0, π/2]. Knowing the properties and domain of arccsc is essential for evaluating limits involving this function.

Behavior of Arccsc(x) as x Approaches Infinity

As x approaches infinity, arccsc(x) approaches the angle whose cosecant is infinitely large, which corresponds to an angle approaching 0 from the positive side. Understanding this behavior allows one to evaluate the limit by relating large values of x to the corresponding angle values.

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)19. lim(x→∞)arccsc(x)