91Ó°ÊÓ

In calculus, limits provide a way to understand the behavior of functions as inputs approach certain values. When we deal with multivariable limits, we're looking at functions with more than one variable—typically functions of the form \( f(x, y) \). This means we want to understand how the function behaves as both \( x \) and \( y \) simultaneously approach specific values. To evaluate a multivariable limit, for example, \( \lim_{(x, y) \to (a, b)} f(x, y) \), we consider the values that the function \( f(x, y) \) tends to as \( (x, y) \) moves closer and closer to \( (a, b) \). It’s crucial to check if the limit exists and is the same from all paths toward \( (a, b) \). If different paths give different limit values, the limit does not exist. However, if the value is consistent, we say that the limit does exist and equals that value. In the exercise, we approached the point \( (1, -3) \), checked the value of the function directly at this point, which is a straightforward approach when the function behaves normally without any holes, asymptotes, or discontinuities.

Evaluating a limit involves a few clear steps, especially for multivariable functions. The process gets more engaging with two-variable functions like \( f(x, y) \). We can:

• Simplify the function where possible, to make the evaluation easier.
• Substitute directly the point values if the function is continuous at the point.
• Consider alternative methods such as using polar coordinates or testing limits along different paths if direct substitution doesn't work.

In the case given, the function \( 3x + 4y - 2 \) is linear, which means it's continuous everywhere. This simplifies our task because we can plug in \( x = 1 \) and \( y = -3 \) directly into the function without any extra steps. Easy, right?Once you substitute the values, you simplify the expression to find the limit: - Substitute \( x = 1 \) and \( y = -3 \).- Calculate: \( 3 \cdot 1 + 4 \cdot (-3) - 2 = -11 \).This method works beautifully for continuous functions, making the evaluation process smooth and straightforward.

Functions of two variables, like \( f(x, y) \), open up a fascinating world beyond what's possible with single-variable functions. These functions depend on two independent variables creating a surface in three-dimensional space. As a result, they allow us to explore multi-dimensional problems and behaviors.These functions have applications in many fields, such as physics, engineering, and economics, where a change in input values along multiple dimensions affects the output. Each value of \( x \) and \( y \) gives a unique \( z \)-value, forming a surface that we often visualize for better understanding. When dealing with limits, functions of two variables require careful consideration since there's more than one path to approach a particular point. In our exercise, the function \( 3x + 4y - 2 \) makes it easier to evaluate as it’s linear, maintaining continuous behavior in every direction and simplifying the task. However, for more complex functions, examining different paths or using specific techniques like continuity tests can be necessary to assert the correct limit value.