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The direct substitution method is often the first approach when evaluating limits. It involves plugging in the limit values directly into the function. If substituting the values does not result in a format that makes mathematical sense, such as dividing by zero, this method won't directly give you the answer.

In our exercise, the function is given by \[ \frac{x^{2}-xy+1}{x^{2}+y^{2}} \]and we're asked about the limit as \((x, y)\) approaches \((1, 0)\).

By substitution, inserting \(x = 1\) and \(y = 0\), gives us:

• \( x^{2} - xy + 1 \rightarrow 1^{2} - 1 \times 0 + 1 = 2 \)
• \( x^{2} + y^{2} \rightarrow 1^{2} + 0^{2} = 1 \)

which results in the clear fraction \(\frac{2}{1} = 2\).

This suggests that, for this limit, the substitution directly guides us to the solution. However, it's always wise to check if the direct substitution method is valid, especially if there's a chance of indeterminate forms.

An indeterminate form arises when substituting the point into the function results in a form like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or other undefined expressions. These forms signal the uncertainty, indicating that simply substituting values won't yield a definitive result.

In our current problem, when we substitute \((x, y) = (1, 0)\) into:

• Numerator: \(x^{2} - xy + 1\) gives us \(2\)
• Denominator: \(x^{2} + y^{2}\) gives us \(1\)

This clear fraction, \(\frac{2}{1}\), indicates no division by zero, nor is there any other type of indeterminate form, like \(\frac{0}{0}\).

Understanding indeterminate forms is crucial, as they guide us towards using more sophisticated techniques, such as L'Hôpital's rule or algebraic manipulation, if needed.

Multivariable limits involve concepts that extend beyond single-variable limits. Here, we are not just looking at a single path approaching a limit point but potentially many paths. The function in question is:\[ \lim _{(x, y) \rightarrow(1,0)} \frac{x^{2}-xy+1}{x^{2}+y^{2}} \]
With multivariable limits, to assert that a limit exists, we must confirm that the same value is approached regardless of the path taken to the limit point.

In the given exercise:

• Substituting directly yielded a simple result: \(2\).
• There were no indeterminate forms indicating consistent behavior across different potential paths, at least around \( x = 1 \) and \( y = 0 \).

Understanding multivariable limits often requires ensuring consistent limit behavior from multiple pathways, essentially confirming that the result doesn't change irrespective of the approach. For more complex problems where direct substitution isn't clear, path analysis becomes essential.