91Ó°ÊÓ

When studying limits in multivariable calculus, we analyze functions with more than one variable. Unlike single-variable calculus, where we approach a limit from two directions, multivariable limits can be approached from infinitely many paths. For instance, in the given exercise, we need to evaluate the limit as \((x, y)\) approaches \( (1, -1) \) in the plane.
The first step involves substituting the values directly into the function. If it results in a determinate form, it shows that the function's limit at that point exists without further simplification. Multivariable limits can sometimes be more complex, requiring techniques like transforming to polar coordinates or using paths such as \((x, y) \to (0, 0) \) along \( y = kx \).

Direct substitution is the simplest method for evaluating a limit. It involves substituting the point into the function directly. If the result is finite and determinate, then this value is the limit. In the given problem, we substitute the values \( x = 1 \) and \( y = -1 \) into the function:
\[ \frac{x^2 - 2xy + y^2}{x-y} \]
This becomes:
\[ \frac{1^2 - 2(1)(-1) + (-1)^2}{1 - (-1)} = \frac{1 + 2 + 1}{2} = 2 \]
Since this is a determinate form, it simplifies to \( 2 \). Thus, the limit exists and equals \( 2 \).

Indeterminate forms occur when direct substitution leads to expressions like \( \frac{0}{0} \), \( \frac{\text{\textinfty}}{\text{\textinfty}} \), \( 0 \times \text{\textinfty} \), and others. These forms require further analysis since the initial result does not provide enough information. Common techniques to resolve indeterminate forms include:

• Simplifying the expression
• Using L'Hôpital's Rule (for limits involving ratios with single variables)
• Factoring
• Using trigonometric identities or series expansions

In our initial direct substitution, the limit \( \frac{x^2 - 2xy + y^2}{x-y} \) at \( (x, y) = (1, -1) \) did not result in an indeterminate form, leading directly to a numerical limit.