The Limit of an Exponential Function Involving Trigonometric Terms

Understanding the behavior of complex functions as they approach infinity is a critical part of advanced calculus. This article delves into finding the limit of (displaystylelim_{x to infty} cos^x frac{pi}{sqrt{x}}), which involves a combination of trigonometric, exponential, and series expansion techniques.

Problem Statement

The challenge is to evaluate the limit:

(y cos^x frac{pi}{sqrt{x}})

Step-by-Step Solution

Step 1: Logarithmic Transformation

First, we apply the natural logarithm to both sides of the equation to simplify the expression:

(ln y ln left(cos^x frac{pi}{sqrt{x}}right))

This can be further simplified using the property of logarithms:

(ln y x ln left(cos frac{pi}{sqrt{x}}right))

Step 2: Taylor Series Expansion

To solve the limit, we use a Taylor series expansion for the cosine function:

(cos frac{pi}{sqrt{x}} 1 - frac{pi^2}{2x} Oleft(frac{1}{x^2}right))

Substituting this into the logarithmic expression:

(ln left(cos frac{pi}{sqrt{x}}right) ln left(1 - frac{pi^2}{2x} Oleft(frac{1}{x^2}right)right))

Using the Taylor series expansion for the natural logarithm near zero:

(ln left(1 uright) u - frac{u^2}{2} O(u^3)) for (u to 0)

We get:

(ln left(cos frac{pi}{sqrt{x}}right) -frac{pi^2}{2x} Oleft(frac{1}{x^2}right))

Step 3: Evaluating the Limit

Substituting the logarithmic expression back into the original equation:

(x ln left(cos frac{pi}{sqrt{x}}right) x left(-frac{pi^2}{2x} Oleft(frac{1}{x^2}right)right))

Further simplification gives:

(x ln left(cos frac{pi}{sqrt{x}}right) -frac{pi^2}{2} Oleft(frac{1}{x}right))

As (x to infty), the term (Oleft(frac{1}{x}right)) approaches zero, so:

(ln y -frac{pi^2}{2})

Therefore:

(y e^{-frac{pi^2}{2}})

Hence:

(displaystylelim_{x to infty} cos^x frac{pi}{sqrt{x}} e^{-frac{pi^2}{2}})

Conclusion

This example demonstrates the power of combining logarithmic and Taylor series expansion techniques to solve complex limits involving trigonometric and exponential functions. Such methods are essential tools in advanced calculus and have applications in various fields, including physics and engineering.

Related Keywords

Webmaster Keywords: trigonometric limits, exponential functions, Taylor series expansion, mathematical limits, advanced calculus, series expansion, logarithmic properties