Problem 62 - Calculus Early Transcendentals

Question

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}+x y-2 y^{2}}{2 x^{2}-x y-y^{2}}$$

Step-by-Step Solution

Answer

The limit is \( 1 \).

Step 1: Factor numerator and denominator

Numerator: \( x^2+xy-2y^2 = (x+2y)(x-y) \).
Denominator: \( 2x^2-xy-y^2 = (2x+y)(x-y) \) (by trial: \( (2x+y)(x-y) = 2x^2-2xy+xy-y^2 \) -- no, that gives \( 2x^2-xy-y^2 \). Correct!).

Step 2: Cancel and evaluate

\( \frac{(x+2y)(x-y)}{(2x+y)(x-y)} = \frac{x+2y}{2x+y} \) for \( x \neq y \). At \( (1,1) \): \( \frac{1+2}{2+1} = 1 \).

Key Concepts

Limits

The concept of limits is a fundamental aspect of calculus. It helps us understand how a function behaves as it approaches a certain point. Limits allow us to evaluate functions that might seem difficult to define or approach values at particular points where they are not directly computable. This can be especially useful in cases involving indeterminate forms like 0/0.

When dealing with two-variable limits, the approach is slightly different. Here, we consider the behavior of the function as both input variables approach specific values. It's important to check if the limit is the same regardless of the path taken to reach that point. Imagine approaching from different directions like along the x-axis, y-axis, or diagonal. Consistency across paths indicates a well-defined limit.

In the exercise provided, the goal is to find the limit of a multivariable expression as \((x, y)\) approaches \((1, 1)\). This involves checking if the limit is the same from various paths, which requires a thorough analysis.

Multivariable functions

Multivariable functions extend the notion of single-variable functions into higher dimensions. Instead of a function with one input and one output, we have functions with multiple inputs and potentially multiple outputs. In mathematical terms, a function \(f(x, y)\) maps pairs of inputs \((x, y)\) to an output value.

Analyzing these functions involves understanding how changes in one variable affect the overall function. It becomes slightly more complex than single-variable calculus since it involves dealing with partial derivatives and gradients. However, the essential tactics remain the same: observe and analyze the behavior of the function as coordinates change.

The exercise at hand considers a function defined by a rational expression in terms of \(x\) and \(y\). The objective is to evaluate this function's behavior as both \(x\) and \(y\) simultaneously approach \(1\). It's crucial to manage each variable's effect on the function, which in this case requires a clever substitution to simplify the problem.

Substitution method

The substitution method is a technique often used in calculus to simplify problems involving limits. The main idea is to replace variables or expressions with easier terms. This can help transform a complex expression into a more manageable one.

In the context of the given problem, using the substitution \(u = x - y\), we transform the multivariable limit problem into a simpler task. This substitution helps to evaluate the original limit by expressing it as a limit of a single variable \(u\).

• The original expression is changed to depend only on \(u\) by rewriting \(x = y + u\).
• The limit then becomes easier to handle since \(u\) tends to 0 as \((x, y)\) approaches \((1, 1)\).

This strategy is incredibly useful when direct evaluation does not yield obvious results or when simplification is necessary to eliminate indeterminate forms.