91Ó°ÊÓ

When evaluating limits in calculus, you may encounter expressions that give rise to indeterminate forms. These occur when direct substitution into an expression results in an undefined expression, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Such forms don't provide clear information about the limit, necessitating further manipulation to resolve it. Consider the task of evaluating the limit \( \lim_{(x, y) \rightarrow (1,1)} \frac{xy - y - 2x + 2}{x-1} \). By substituting \( x=1 \) and \( y=1 \), we find that the expression results in \( \frac{0}{0} \), an indeterminate form. This indicates that more work is needed to discover the true behavior of the function as \( (x, y) \) approaches \( (1, 1) \). Techniques such as factoring or rationalizing are typically used to eliminate indeterminacy.

Rational expressions are fractions where both the numerator and the denominator are polynomials. Such expressions are straightforward to handle, but they become challenging when the denominator has the potential to be zero, especially in limit problems. In the expression \( \frac{xy - y - 2x + 2}{x-1} \), the presence of \( x-1 \) in the denominator makes it a rational expression that requires careful treatment for limit evaluation. The potential problem occurs because if \( x=1 \) is plugged directly, it results in a zero in the denominator leading to undefined behavior or an indeterminate form. To manage this, we often must rewrite or simplify the expression, using algebraic manipulations like factorization or cancelling out common terms once they appear.

Factorization is a powerful tool for simplifying expressions in calculus, especially when dealing with indeterminate forms or rational expressions. In our example, factorization helps to rewrite the numerator \( xy - y - 2x + 2 \). By grouping, notice that you can express it as \( x(y-2) - 1(y-2) \). This can be further rewritten using factorization as \((x-1)(y-2) \). The beauty of factorization is that it can transform seemingly complex expressions into simpler ones that cancel out problematic terms in the denominator. Here, \( x-1 \) is a common factor in both the numerator and denominator, allowing it to cancel and simplify the whole expression. Once simplified, the evaluation of limits becomes much more direct and manageable.

Simplification plays an essential role in calculus, especially for evaluating limits. Simplifying expressions often involves cancelling terms, factorizing, or using trigonometric identities. In the problem \( \lim_{(x, y) \rightarrow (1,1)} \frac{(x-1)(y-2)}{x-1} \), the simplification process allowed us to remove the indeterminacy by cancelling the common \( x-1 \) term from both the numerator and denominator. The expression then reduces to \( y-2 \), which no longer poses a problem as \( x \) approaches 1. When simplification is successfully applied, it effectively reveals the "true" nature of the expression, making limit calculation straightforward and avoiding undefined behavior. As a result, substituting \( y=1 \) easily gives the final limit as \(-1\).