Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\lim _{(x, y) \rightarrow(1,2)} \frac{x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}{y-2 x^{2}}\)

Step-by-Step Solution

Answer

The limit is undefined at (1, 2) as the denominator equals zero.

Step 1: Analyze the Limit

First, we need to determine if the limit is of an indeterminate form. Substitute the point \(x=1, y=2\) into the expression to check if both the numerator and denominator approach zero as \(x ightarrow 1\) and \(y ightarrow 2\). We have \(1^3 - 3(1)^2(2) + 3(1)(2)^2 - 2^3 = 1 - 6 + 12 - 8 = -1\) and \(2 - 2(1)^2 = 0\). Although the numerator does not approach zero at \(x=1, y=2\), the denominator does. Thus, the expression does not result in a \(\frac{0}{0}\) indeterminate form. Hence, we proceed with direct evaluation of the limit.

Step 2: Direct Substitution

Since the limit does not result in an indeterminate form \(\frac{0}{0}\), directly substitute \(x=1\) and \(y=2\) into the expression. Compute the numerator \(1^3 - 3(1)^2(2) + 3(1)(2)^2 - 2^3 = -1\), and the denominator \(2 - 2(1)^2 = 0\). The division of \(-1\) by \(0\) indicates an undefined result, suggesting that the function approaches negative infinity or positive infinity or does not have a limit at this point.

Key Concepts

Understanding Indeterminate Forms

In calculus, an indeterminate form is a mathematical expression that initially appears to fizzle out into forms like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), among others, during limit evaluation. Such forms require deep inspection because they don't immediately reveal the limit's behavior as variables approach specific values.

• Indeterminate forms occur when both the numerator and denominator of a rational expression approach zero; this is called a \(\frac{0}{0}\) form.
• They can also arise in cases involving infinity, such as \(\frac{\infty}{\infty}\) or differences like \(\infty - \infty\).

Recognizing indeterminate forms is crucial because it informs us about the necessity to apply techniques like L'Hopital's rule, algebraic manipulation, or sometimes even special limits theorems, to resolve the apparent indeterminacy. In our exercise, the expression did not simplify to a \(\frac{0}{0}\) form, which allows us to explore other options including direct substitution.

Using Direct Substitution When Possible

Direct substitution is a straightforward method to evaluate limits. This technique involves plugging in the specified values for the variables, assuming they do not lead to indeterminate forms like \(\frac{0}{0}\).

• If direct substitution gives you a defined number, then that number is the limit of the function at the point of interest.
• However, if substitution leads to undefined values like \(\frac{a}{0}\) (with \(a eq 0\)), further analysis is needed to understand the behavior near that point.

In our exercise, substituting \(x = 1\) and \(y = 2\) into the function does not encounter a \(\frac{0}{0}\) form. The numerator evaluates to \(-1\) and the denominator to zero, suggesting a division by zero scenario which means the function could be approaching infinity. Calculating these values helps us understand the potential for the function to not have a finite limit.

Exploring Limit Analysis for Comprehensive Understanding

Limit analysis refers to examining a function's behavior as it approaches a specific point, using various techniques to discover whether a limit exists and, if so, to calculate it. This analysis often involves:

• Checking for continuity at the point of interest.
• Considering different paths in multivariable limits to determine if the limit changes based on the path taken.
• Using epsilon-delta definitions for rigorous proof in theoretical situations.

In multivariable calculus, paths matter because a function might have different limits from various directions around a point. In our case, even if direct substitution shows an undefined form, exploring other approaches could reveal more about the function's trending direction, like towards positive or negative infinity, further signifying that no definitive limit exists at the point \((1, 2)\). By assessing multiple aspects, we get a clearer picture of the function's behavior and limits in true multivariable context.