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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