Evaluating The Limit Of (1 + √2 Sin Θ) / Cos 2θ As Θ Approaches -π/4

Introduction

In the realm of calculus, evaluating limits is a fundamental concept, serving as the bedrock for understanding continuity, derivatives, and integrals. Among the diverse array of limit problems, some present unique challenges that demand a blend of trigonometric identities, algebraic manipulation, and a keen understanding of limit properties. In this comprehensive exploration, we delve into the intricacies of evaluating the limit:

$$\lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1+\sqrt{2} \sin \theta}{\cos 2 \theta}$$

This particular limit, at first glance, appears to be a straightforward substitution problem. However, a direct substitution of ${\theta = -\frac{\pi}{4}}$ leads to an indeterminate form, specifically ${\frac{0}{0}}$. This signals the need for a more sophisticated approach, one that involves manipulating the expression to eliminate the indeterminate form and reveal the true behavior of the function as ${\theta}$ approaches ${-\frac{\pi}{4}}$. Our journey will involve employing trigonometric identities, algebraic techniques, and a careful analysis of the function's behavior near the point of interest. We'll navigate through the steps with meticulous detail, providing explanations and insights along the way, ensuring a clear and comprehensive understanding of the solution.

The Challenge of Indeterminate Forms

When we encounter limits, the initial approach is often direct substitution. We simply plug in the value that the variable is approaching and see what we get. However, in some cases, this direct substitution leads to what we call indeterminate forms. These forms, such as ${\frac{0}{0}}$, ${\frac{\infty}{\infty}}$, ${0 \cdot \infty}$, and others, don't give us a clear answer about the limit's value. They tell us that we need to do more work to evaluate the limit.

The indeterminate form ${\frac{0}{0}}$ arises when both the numerator and the denominator of a fraction approach zero as the variable approaches a certain value. In our case, as ${\theta}$ approaches ${-\frac{\pi}{4}}$, both ${1 + \sqrt{2} \sin \theta}$ and ${\cos 2\theta}$ approach zero. This is because ${\sin(-\frac{\pi}{4}) = -\frac{1}{\sqrt{2}}}$ and (\cos(2(-\frac{\pi}{4})) = \cos(-\frac{\pi}{2}) = 0). The presence of this indeterminate form signifies that a simple substitution is insufficient, and we must employ other techniques to unveil the limit's true value. This is where the power of trigonometric identities and algebraic manipulation comes into play, allowing us to transform the expression into a form where the limit can be evaluated directly.

Navigating the Indeterminate Form: A Strategic Approach

To effectively tackle the indeterminate form ${\frac{0}{0}}$ in our limit problem, we need a strategic approach. This involves identifying the problematic parts of the expression and employing techniques to simplify or transform them. In our case, the presence of trigonometric functions suggests that trigonometric identities might be helpful. The expression ${\cos 2\theta}$ in the denominator is a prime candidate for simplification using the double-angle formula. Additionally, the term ${1 + \sqrt{2} \sin \theta}$ in the numerator warrants careful consideration, as it is directly contributing to the zero value when ${\theta = -\frac{\pi}{4}}$.

Our strategy will involve the following steps:

1. Apply the double-angle formula for cosine: We will replace ${\cos 2\theta}$ with a suitable equivalent form, such as ${1 - 2\sin^2 \theta}$ or ${\cos^2 \theta - \sin^2 \theta}$. The choice of which form to use will depend on our goal of simplifying the expression and eliminating the indeterminate form.
2. Algebraic manipulation: After applying the trigonometric identity, we will look for opportunities to factor, simplify, or combine terms. This might involve factoring the numerator or denominator, or multiplying the expression by a clever form of 1 to rationalize or simplify.
3. Evaluate the limit: Once we have simplified the expression, we will re-evaluate the limit using direct substitution. If the indeterminate form has been eliminated, we should be able to obtain a finite value for the limit. If the indeterminate form persists, we may need to repeat the process or employ other techniques, such as L'Hôpital's Rule (though we will focus on trigonometric and algebraic methods in this exploration).

Unveiling the Solution: Step-by-Step Evaluation

Let's embark on the journey of solving the limit step-by-step. Following our strategic approach, the first step involves applying a trigonometric identity to the denominator, ${\cos 2\theta}$. We have several choices for the double-angle formula, but the form ${\cos 2\theta = 1 - 2\sin^2 \theta}$ appears most promising in this context, as it introduces a ${\sin^2 \theta}$ term, which relates to the ${\sin \theta}$ term in the numerator. Substituting this identity, our limit becomes:

$$\lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1+\sqrt{2} \sin \theta}{1 - 2\sin^2 \theta}$$

Now, the denominator is a quadratic expression in terms of ${\sin \theta}$. This suggests that we might be able to factor it. Indeed, we can rewrite the denominator as a difference of squares, or further manipulate to facilitate cancellation with terms in the numerator. Let's proceed with factoring the denominator:

$$1 - 2\sin^2 \theta = (1 - \sqrt{2}\sin \theta)(1 + \sqrt{2}\sin \theta)$$

Notice that we now have a factor of ${(1 + \sqrt{2}\sin \theta)}$ in both the numerator and the denominator. This is a crucial step, as it allows us to cancel out the term that is causing the indeterminate form. Cancelling this common factor, we get:

$$\lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1}{1 - \sqrt{2}\sin \theta}$$

Now, we have a simplified expression. Let's try direct substitution again. Plugging in ${\theta = -\frac{\pi}{4}}$, we get:

$$\frac{1}{1 - \sqrt{2}\sin(-\frac{\pi}{4})} = \frac{1}{1 - \sqrt{2}(-\frac{1}{\sqrt{2}})} = \frac{1}{1 + 1} = \frac{1}{2}$$

Thus, we have successfully evaluated the limit. The indeterminate form has been eliminated, and we have arrived at a finite value.

Conclusion: The Power of Strategic Manipulation

In conclusion, the limit

$$\lim _{\theta \rightarrow-\frac{\pi}{4}} \frac{1+\sqrt{2} \sin \theta}{\cos 2 \theta}$$

presents a classic example of a limit problem that requires more than just direct substitution. The initial indeterminate form ${\frac{0}{0}}$ necessitates a strategic approach involving trigonometric identities and algebraic manipulation. By skillfully applying the double-angle formula for cosine and factoring the resulting expression, we were able to identify and cancel the problematic term, ultimately leading to a simplified expression that could be evaluated directly.

The key takeaway from this exploration is the importance of strategic manipulation in evaluating limits. When faced with an indeterminate form, it is crucial to identify the underlying cause and employ appropriate techniques to eliminate it. Trigonometric identities, algebraic factorization, and other tools in our mathematical arsenal serve as powerful aids in this endeavor. Moreover, this example underscores the interconnectedness of different mathematical concepts, highlighting how trigonometry and algebra work in concert to solve calculus problems. The solution, ${\frac{1}{2}}$, not only provides the answer but also reinforces the elegance and power of mathematical reasoning.

By understanding these principles and practicing problem-solving techniques, we can confidently navigate the world of limits and derivatives, unlocking the deeper insights that calculus offers into the behavior of functions and the world around us. The journey through this problem serves as a valuable stepping stone towards mastering the broader landscape of calculus and its applications.