Evaluating The Trigonometric Limit As X Approaches Pi

Introduction

In the realm of calculus, evaluating limits is a fundamental concept. Limits help us understand the behavior of functions as their input approaches a specific value. This article delves into the evaluation of a complex trigonometric limit, providing a step-by-step guide to simplify the expression and arrive at the solution. We will explore the limit of the following expression as x approaches π:

[ 1 - sin²(x/2) * (cos(x/4) + sin(x/4)) ] / [ cos(x/2) * (cos²(x/4) - sin²(x/4)) ]

This expression involves trigonometric functions, fractions, and a limit approaching π. To tackle this, we will employ trigonometric identities, algebraic manipulations, and a keen understanding of limit properties. Let's embark on this journey of mathematical exploration and unravel the intricacies of this limit.

Breaking Down the Expression

To begin, we need to simplify the given trigonometric expression. The key to unlocking this limit lies in recognizing and applying trigonometric identities effectively. The expression we are dealing with is:

[ 1 - sin²(x/2) * (cos(x/4) + sin(x/4)) ] / [ cos(x/2) * (cos²(x/4) - sin²(x/4)) ]

Let's break down the numerator and the denominator separately to identify potential simplifications. In the numerator, we have 1 - sin²(x/2), which immediately suggests the Pythagorean trigonometric identity: sin²θ + cos²θ = 1. This can be rearranged to cos²θ = 1 - sin²θ. Applying this identity to our expression, we can replace 1 - sin²(x/2) with cos²(x/2). This substitution is a crucial first step in simplifying the overall expression. The numerator now becomes:

cos²(x/2) * (cos(x/4) + sin(x/4))

Moving on to the denominator, we encounter cos²(x/4) - sin²(x/4). This expression is a direct application of another fundamental trigonometric identity: the double-angle formula for cosine. Specifically, cos(2θ) = cos²θ - sin²θ. In our case, θ is x/4, so 2θ becomes x/2. Thus, we can replace cos²(x/4) - sin²(x/4) with cos(x/2). The denominator now simplifies to:

cos(x/2) * cos(x/2) = cos²(x/2)

By applying these trigonometric identities, we've transformed the original complex expression into a more manageable form. This simplification is vital for evaluating the limit as x approaches π. The rewritten expression is:

[ cos²(x/2) * (cos(x/4) + sin(x/4)) ] / [ cos²(x/2) ]

Simplifying the Expression Further

With the trigonometric identities applied, the expression has become considerably simpler. We now have:

[ cos²(x/2) * (cos(x/4) + sin(x/4)) ] / [ cos²(x/2) ]

Observe that cos²(x/2) appears in both the numerator and the denominator. This allows us to cancel out this common factor, further simplifying the expression. However, it's crucial to acknowledge that this cancellation is valid only when cos²(x/2) is not equal to zero. We'll address this condition later when we discuss the limit as x approaches π.

Assuming cos²(x/2) is not zero, we can cancel it from the numerator and the denominator. This leaves us with:

cos(x/4) + sin(x/4)

This simplified expression is significantly easier to work with compared to the original one. Now, we can directly evaluate the limit as x approaches π. The process of simplifying the expression has made the limit evaluation much more straightforward.

Evaluating the Limit as x Approaches π

Having simplified the original expression to cos(x/4) + sin(x/4), we are now in a position to evaluate the limit as x approaches π. This involves substituting π for x in the simplified expression and calculating the result.

So, we have:

lim (x→π) [cos(x/4) + sin(x/4)] = cos(π/4) + sin(π/4)

To evaluate this, we need to recall the values of cosine and sine at π/4 radians (or 45 degrees). From the unit circle or trigonometric tables, we know that:

cos(π/4) = √2 / 2
sin(π/4) = √2 / 2

Substituting these values into our expression, we get:

√2 / 2 + √2 / 2

This simplifies to:

2 * (√2 / 2) = √2

Therefore, the limit of the expression as x approaches π is √2. This result highlights the power of simplification techniques in evaluating limits. By applying trigonometric identities and algebraic manipulations, we transformed a complex expression into a simple one, making the limit evaluation straightforward.

Addressing the Cancellation Condition

Earlier, we canceled out the cos²(x/2) term in the expression. However, we noted that this cancellation is valid only if cos²(x/2) is not equal to zero. Now, we need to verify that this condition holds as x approaches π.

Let's consider cos(x/2) as x approaches π. We have:

cos(π/2)

From the unit circle or trigonometric knowledge, we know that cos(π/2) = 0. Therefore, cos²(π/2) is also 0.

This raises a crucial point: we canceled a term that becomes zero at the limit point. This situation often requires a more rigorous analysis to ensure the validity of the cancellation. In this case, we can examine the behavior of the expression as x gets arbitrarily close to π, but not exactly equal to π.

As x approaches π, but is not equal to π, x/2 approaches π/2, but is not equal to π/2. Thus, cos(x/2) approaches 0, but is not exactly 0. Similarly, cos²(x/2) approaches 0, but is not exactly 0. This means that for values of x close to π (but not equal to π), we can indeed cancel the cos²(x/2) term.

The fact that the limit exists and is equal to √2 despite the cancellation condition reinforces the validity of our approach. The limit captures the behavior of the function as it gets infinitesimally close to x = π, even though the function itself might be undefined at x = π due to the zero in the denominator.

Conclusion

In this article, we successfully evaluated the limit of a complex trigonometric expression as x approaches π. We began by simplifying the expression using trigonometric identities, specifically the Pythagorean identity and the double-angle formula for cosine. This simplification was crucial in transforming the expression into a manageable form.

We then canceled a common factor, cos²(x/2), from the numerator and the denominator, which further simplified the expression. However, we also addressed the condition under which this cancellation is valid, ensuring the rigor of our approach.

Finally, we evaluated the limit of the simplified expression by substituting π for x and calculating the result. The limit of the expression as x approaches π was found to be √2.

This exercise highlights the importance of trigonometric identities, algebraic manipulations, and a careful consideration of limit properties in evaluating complex limits. The ability to simplify expressions and address potential issues like division by zero is essential for success in calculus and related fields. By mastering these techniques, we can confidently tackle a wide range of limit problems and gain a deeper understanding of the behavior of functions.