Calculating the Limit of Trigonometric Expressions Involving Products of Sine and Cosine Functions

Introduction

Understanding the behavior of trigonometric functions as they approach certain points is a fundamental aspect of calculus. In this article, we will explore the calculation of a specific limit involving products of sine and cosine functions. We will start by providing a detailed explanation and then present an alternate solution for a similar but distinct problem.

Solving the Limit: limx→0 [ 1 - cos√x - ... - cos√nx ] / [ sinx ... sinnx ] using Trigonometric Identities

In this limit problem, we need to evaluate limx→0 [ 1 - cos√x - ... - cos√nx ] / [ sinx ... sinnx ]. We will make use of trigonometric identities and properties of limits to find the solution.

First, we will simplify each individual cosine term using the identity: cosθ 1 - 2sin2θ/2. Applying this to each cosine term, we get:

1 - cos√x 2sin2√x/2

1 - cos√nx 2sin2√nx/2

Next, we use the sum-to-product identities for sine terms:

sinx ... sinnx sinnx/2 { 2cosnx/2 2cos(n-2)x/2 ... 2cosx/2 }

Combining and simplifying the expression, we get:

limx→0 [ 2sin2√x/2 ... 2sin2√nx/2 ] / [ sinnx/2 { 2cosnx/2 2cos(n-2)x/2 ... 2cosx/2 } ]

Applying limit properties and the approximation sinθ ≈ θ for small θ, we can simplify the expression again to:

limx→0 [ √x ... √nx / nx/2 ] 2/n

Therefore, the limit converges to 2/n.

Alternate Problem: The Limit Doesn’t Exist for a Particular Case

It is important to note that the limit doesn’t exist in a particular scenario. For example, if we consider the limit of the expression limx→0 (1 - cossqrt{x} ... cossqrt{nx}) / (sinx ... sin{nx}), the numerator approaches 1 - n, while the denominator approaches 0. This indicates that the limit does not exist, as the denominator can approach 0 through negative and positive values from the left and right sides, respectively.

However, if the original problem was meant to be a different expression, such as limx→0 (1 - cos(sqrt{x}) ... cos(sqrt{nx})) / (sin(x) ... sin(nx)), the solution would be different. Here, we can observe that:

cos(sqrt{kx}) 1 - kx/2 O(x2)

Using this, we can simplify the product of cosine terms to:

cossqrt{x}cossqrt{2x} ... cossqrt{nx} 1 - x(1 2 ... n)/2 O(x2) 1 - nx/2 O(x2)

Similarly, for the product of sine terms:

sinxsin2x ... sin{nx} 1 x(1 2 ... n)/2 O(x2) 1 nx/2 O(x2)

Thus, the limit is:

(1 - nx/2 O(x2)) / (1 nx/2 O(x2)) ≈ 1/2 as x → 0

Therefore, the limit converges to 1/2 in this case.

Conclusion

In this article, we have explored the calculation of two distinct but related limits involving products of sine and cosine functions. The first problem led us to a limit of 2/n, while the second demonstrated that the limit could be 1/2. Understanding these calculations can be valuable in various mathematical and analytical contexts, especially in the field of calculus and trigonometry.