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When calculating limits, an indeterminate form presents a situation where substitution of the given values results in an undefined expression, such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or more. These forms signal that direct evaluation isn't possible, and some algebraic manipulation or adjustment is needed to evaluate the limit correctly. Indeterminate forms show up frequently in multivariable limits, where more than one variable approaches a specific set of values simultaneously.

In the exercise provided, substituting the values \((1, 1, 1)\) into the expression \(\frac{x^{2}+xy-xz-yz}{x-z}\) results in \(\frac{0}{0}\), an indeterminate form. This type requires us to delve deeper into the algebraic structure of the expression to find a pattern or simplification that resolves the form, allowing us to find the actual limit.

Limit evaluation in the context of multivariable calculus involves determining the behavior of a function as its variables approach certain points. These evaluations must account for all variables involved, making them often more complex than single-variable limits.

• Begin by substituting the values directly into the function.
• Identify if the result leads to an indeterminate form.
• Simplify the function to resolve the indeterminacy and make it evaluable.

In our original exercise, the objective was to evaluate the limit of a multivariable function, which initially presented as \(\frac{0}{0}\). Evaluating limits like these often necessitates factoring or other algebraic techniques to simplify the expression into one that is not indeterminate and can be more easily evaluated through substitution.

Simplifying a function to evaluate its limit is crucial, especially when dealing with indeterminate forms. The simplification process often involves algebraic techniques such as factoring, canceling common terms, or rearranging parts of the expression.

In our specific example, simplifying \(\frac{x^{2}+xy-xz-yz}{x-z}\) involved factoring the numerator. The original terms in the numerator were restructured using common factoring to expose the \((x-z)\) factor in both the numerator and denominator. This allowed for canceling \((x-z)\), leaving behind a simplified expression: \(x + y - z\).

This stepwise simplification provided a straightforward path to substitute back the values \((1, 1, 1)\) for \((x, y, z)\), leading to the evaluation of the simplified expression, resulting in the final limit. Understanding how to simplify correctly can turn an indeterminate or complex expression into one that is easy to handle and solve.