The Limit of the Ratio of Hyperbolic Sine to Sine Function as x Approaches Zero

In mathematics, particularly in calculus, the limits of certain functions as they approach a specific value are crucial concepts. One common and interesting example involves the limit of the ratio of the hyperbolic sine function sinh(x) to the sine function sin(x) as x approaches zero. This article explores the value of this limit and the underlying mathematical principles, including the application of L'H?pital's rule.

Understanding the Functions

The hyperbolic sine function, sinh(x), and the sine function, sin(x), are important trigonometric functions with distinct properties. The hyperbolic sine is defined as:

sinh(x) (ex - e-x) / 2

While the sine function is defined as:

sin(x)

Applying L'H?pital's Rule

The concept of limits can sometimes yield indeterminate forms, such as 0/0 or ∞/∞. In these cases, L'H?pital's rule can be applied, which states that for two differentiable functions fx and gx, the limit of the ratio of these functions as x approaches a certain value can be determined by the limit of the ratio of their derivatives.

Solution of the Limit Problem

Let's consider the problem:

limx→0 [sinh(x) / sin(x)]

Using L'H?pital's rule, we first differentiate the numerator and the denominator:

Numerator:

First, we find the derivative of sinh(x): f'(x) cosh(x)

Denominator:

Next, we differentiate the sine function:" "

Let's differentiate the denominator:

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g'(x) cos(x)

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Now, we apply L'H?pital's rule:

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limx→0 [sinh(x) / sin(x)] limx→0 [cosh(x) / cos(x)]

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Evaluating the limit at x 0:

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cosh(0) / cos(0) 1 / 1 1

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Hence:

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limx→0 [sinh(x) / sin(x)] 1

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Mathematical Insight

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This result, while simple, highlights the behavior of the hyperbolic and trigonometric functions near zero. It demonstrates how hyperbolic functions, which are defined based on the exponential function, can be closely related to trigonometric functions, especially around the origin.

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Conclusion

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The limit of the ratio of the hyperbolic sine to the sine function as x approaches zero is a fascinating and instructive concept in calculus. Understanding such limits provides deeper insight into the behavior of different functions and their interrelations, which is valuable for solving more complex problems in mathematics and related fields.

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Further Reading

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For those interested in delving deeper into these topics, exploring the properties of both hyperbolic and trigonometric functions, and the applications of L'H?pital's rule, there are several resources available. These include advanced calculus textbooks, online tutorials, and specific articles on mathematical limits.

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Key Keywords

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hyperbolic sine, sine function, limit