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Determine the infinite limit. $ \displaystyle \lim_{x \to 2\pi^-}x\csc x $

   Determine the infinite limit.

$ \displaystyle \lim_{x \to 2\pi^-}x\csc x $

Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 2, Problem 39

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Solved by Expert Ma. Theresa Alin

08/26/2021

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So, we can rewrite the limit as: \[ \lim_{x \to 2\pi^-}x\csc x = \lim_{x \to 2\pi^-} \frac{x}{\sin x} \] Show more…

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Determine the infinite limit. $ \displaystyle \lim_{x \to 2\pi^-}x\csc x $

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Key Concepts

Asymptotic Analysis

Asymptotic analysis involves studying the behavior of functions as the variable approaches a specific value where the function may become unbounded, such as near a vertical asymptote. This technique is used to determine how rapidly functions grow larger and whether they tend to infinity or negative infinity.

Cosecant Function

The cosecant function is defined as the reciprocal of the sine function. Its behavior is closely tied to that of sine; particularly, it encounters discontinuities and tends toward infinity when the sine of the angle is zero. Understanding this reciprocal relationship is key when analyzing limits involving the cosecant function.

One-Sided Limit

A one-sided limit examines the behavior of a function as the input approaches a specific point from one direction (either from the left or from the right). This concept is crucial when a function behaves differently depending on the direction of approach, such as near discontinuities or boundaries of its domain.

Limit

In calculus, a limit describes the behavior of a function as the input approaches a particular value. It is fundamental for understanding continuity, derivatives, and integrals. Analyzing limits helps determine if a function approaches a finite value, diverges to infinity, or oscillates without settling.

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