Problem 20 - Calculus of a Single Variable

Question

In Exercises 17–22, find the limit. $$ \lim _{x \rightarrow-\infty} \operatorname{csch} x $$

Step-by-Step Solution

Answer

The limit as x approaches negative infinity of csch(x) is 0.

Step 1: Recall the definition of csch

The hyperbolic cosecant function, csch, is defined as the reciprocal of the sinh function. Therefore, \(csch(x) = \frac{1}{sinh(x)}\).

Step 2: Apply the definition

Substitute the definition of csch into the limit. The limit becomes \( \lim _{x \rightarrow-\infty} \frac{1}{sinh(x)} \).

Step 3: Evaluate the limit

As x approaches negative infinity, \(sinh(x)\) approaches negative infinity. Therefore, the limit becomes zero as the denominator becomes infinitely large.

Key Concepts

Hyperbolic Cosecant

When we encounter hyperbolic functions, understanding their definitions and properties is crucial. The hyperbolic cosecant, denoted as \(csch(x)\), is defined by the reciprocal of the hyperbolic sine function \(sinh(x)\). This means that \(csch(x) = \frac{1}{sinh(x)}\).

Hyperbolic functions like \(csch(x)\) are similar to trigonometric functions but are related to hyperbolas rather than circles. The hyperbolic sine function from which \(csch(x)\) is derived can be expressed using exponential functions as \(sinh(x) = \frac{e^x - e^{-x}}{2}\), allowing for the exploration of properties that are not immediately apparent from its definition as a reciprocal.

Evaluating Limits at Infinity

Evaluating limits at infinity involves determining what value a function approaches as the input grows without bound, either positively or negatively. When we take the limit as \(x\) approaches negative infinity for \(csch(x)\), it fundamentally deals with the behavior of the hyperbolic function as its argument becomes very large in the negative direction.

To do this properly, it's helpful to think about the behavior of the exponential functions that make up \(sinh(x)\) as \(x\) becomes very negative. One part of the function, \(e^x\), approaches zero, and the other, \(e^{-x}\), grows very large. As a result, \(sinh(x)\) itself becomes very large in the negative direction. Since \(csch(x)\) is the reciprocal of \(sinh(x)\), then as \(sinh(x)\) grows, \(csch(x)\) approaches zero.

Hyperbolic Functions

Hyperbolic functions, including hyperbolic sine (\(sinh\)), hyperbolic cosine (\(cosh\)), and hyperbolic cosecant (\(csch\)), are analogs of the common trigonometric functions but for a hyperbola. They are defined using exponential functions, and like their trigonometric counterparts, they have specific identities and properties.

For example, the hyperbolic functions satisfy identities similar to trigonometric identities. One such is the hyperbolic Pythagorean identity: \(cosh^2(x) - sinh^2(x) = 1\). Though these functions may appear esoteric at first glance, they are incredibly useful in various areas including calculus, physics, and engineering, particularly when dealing with scenarios that exhibit hyperbolic geometry or growth patterns that follow hyperbolic trajectories, such as certain rates of decay or acceleration.