91Ó°ÊÓ

In calculus, factoring expressions is a key technique to simplify limits. When faced with a complex expression like \(xy - y - 2x + 2\), it's often beneficial to rearrange and group terms strategically for factoring. This can help reveal hidden structures within the expression.To factor by grouping, you first look for common factors in different sets of terms. In our expression:

• Group the first two terms: \(xy - y\) gives us \(y(x-1)\).
• Group the last two terms: \(-2x + 2\) equates to \(-2(x-1)\).

This method relies on recognizing patterns that allow the expression to be rewritten in a product of simpler factors.
Thus, by carefully rearranging and grouping, we've factored the entire expression as \((x-1)(y-2)\).Factoring like this becomes crucial as part of the simplifying process, particularly when preparing to cancel terms and evaluate limits.

When dealing with limits that involve fractions, simplifying them can often make the limit easier to evaluate. For the expression \(\frac{(x-1)(y-2)}{x-1}\), factoring allowed us to see a common term \((x-1)\) in both the numerator and the denominator.Simplifying the fraction involves:

• Cancelling the common term \((x-1)\) from the numerator and denominator. This step drastically simplifies the expression.
• Ensuring that during this cancellation, the original condition \(x eq 1\) is preserved. This condition is critical to avoid division by zero.

Being able to simplify expressions in this way helps reveal the core behavior of the function as it approaches the limits.
In this case, after cancellation, we are left with the simpler expression \(y - 2\). By simplifying the fraction, the task of computing the limit becomes straightforward.

Evaluating two-variable limits involves understanding how a function behaves as two variables simultaneously approach specific values. In our problem, both \(x\) and \(y\) are headed towards the point \((1,1)\).Steps to evaluate the limit include:

• First, simplifying the expression where possible, as we did with \(y-2\) after canceling \((x-1)\).
• Substituting the limit point values into the simplified expression. Here, substituting \(y = 1\) gives us \(1 - 2 = -1\).

It's important to double-check that the conditions (such as \(x eq 1\)) hold true throughout. This ensures we're evaluating the limit properly.

Two-variable limits can often seem daunting, but breaking them down through simplification and careful substitution can reveal accurate insights about the function's behavior as it nears specific points.