91Ó°ÊÓ

Multivariable limits extend the concept of limits, which we are familiar with from single-variable calculus, to functions of more than one variable. In the exercise above, we deal with a function of two variables, \(x\) and \(y\). As both core to understanding this topic is that both variables are allowed to "move" simultaneously, approaching a certain point. Here, it's \((1, \frac{\pi}{6})\). The challenge with multivariable limits is determining if the function approaches a single value regardless of the path taken through the coordinate plane to approach this point. If the limit exists, then it does so for all paths leading to the target point.

When evaluating multivariable limits:

• Consider substituting the variables directly with the approaching values if the function is continuous at that point.
• If direct substitution leads to an indeterminate form, explore paths like lines or curves approaching the limit point.
• Explore whether different paths lead to the same limit, thus confirming the existence of the limit.

Understanding these approaches provides a comprehensive view of how multivariable functions behave around certain points.

Trigonometric limits are limits that involve trigonometric functions, such as \(\sin\), \(\cos\), and \(\tan\). In the given problem, the trigonometric function is \(\sin y\). Evaluating trigonometric functions at specific angles often simplifies computations because these functions have known values for angles at right triangles. For example, \(\sin(\frac{\pi}{6}) = \frac{1}{2}\).

Key points to consider when dealing with trigonometric limits include:

• Make use of trigonometric identities, which can simplify the analysis of the limits.
• Understand reference angles and their corresponding coordinates on the unit circle.
• Apply L'Hôpital's Rule if necessary for indeterminate forms involving trigonometric functions.

This approach simplifies computations and aids in more efficiently reaching the solution.

Evaluating limits systematically is vital to getting the correct solution. The step-by-step methodology helps manage potentially complex expressions with ease. For this problem, each step led smoothly into the next. Here's a breakdown of an effective strategy:

1. **Identify the Problem**: Understanding what the limit involves is crucial. Ask yourself if it's directly substitutable or requires more work, like examining paths or using rules.2. **Substitute If Appropriate**: If the function is continuous at the point, substitute the variables with approaching values. Avoid substitution in cases that create division by zero or other undefined forms.3. **Evaluate Expressions**: Simplify arithmetic and trigonometric evaluations. Recognize common trigonometric evaluations, like \(\sin(\frac{\pi}{6})\), which equal \(\frac{1}{2}\).By following these steps, one ensures a structured approach to solving limit problems. A clear understanding of each step helps in tackling even more complex multivariable limit questions smoothly.