Problem 63 - Calculus 3

Question

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,0) 4 x^{2}-10 y^{2}+6} $$

Step-by-Step Solution

Answer

The limit exists and is 6.

Step 1: Plug the limit point into the expression

Evaluate the function by substituting the values of \(x\) and \(y\) with 0, as the limit approaches \((0, 0)\). The function is \(4x^2 - 10y^2 + 6\). Substitute \(x = 0\) and \(y = 0\), which gives us \(4(0)^2 - 10(0)^2 + 6\).

Step 2: Simplify the expression

Calculate the expression obtained after substitution. First, compute \(4(0)^2\), which is 0, and \(-10(0)^2\), which is also 0. So the expression simplifies to \(0 + 0 + 6\).

Step 3: Analyze the simplified expression

The simplified form of the expression from Step 2 is \(6\). This indicates that regardless of how \((x, y)\) approaches \((0, 0)\), the limit of the function remains 6.

Key Concepts

Evaluating Limits

Evaluating limits, especially in multivariable calculus, is about finding what value a function approaches as the input points get closer to a certain point. Typically, this point is denoted as \((a, b)\) in two-dimensional space. In our problem, we want to determine what happens as \((x, y)\) approaches \((0, 0)\) for the function \(4x^2 - 10y^2 + 6\).

Here's how to evaluate the limit step-by-step:

• **Substitute the values:** Begin by substituting \(x = 0\) and \(y = 0\) into the function. For our function, this involves replacing those variables directly with zero.
• **Simplify the expression:** After substitution, any terms involving \(x\) or \(y\) will become zero. So, you calculate \(4(0)^2 - 10(0)^2 + 6 = 6\).
• **Conclude the result:** Since after substitution and simplification the limit gives a constant number (6), the evaluated limit is 6, representing the value the function approaches.

There is no need for further methods if substitution works, confirming that this function approaches the constant value indeed.

Multivariable Functions

Multivariable functions are expressions that involve more than one variable. In simple terms, these functions depend on two or more inputs to produce a single output. This complexity allows them to describe a vast array of real-world situations.

For instance, consider our function \(f(x, y) = 4x^2 - 10y^2 + 6\). Here, it depends on the variables \(x\) and \(y\). The surface or graph of such a function would exist in three-dimensional space as opposed to a single line in two-dimensional space.

• **Representation:** Multivariable functions are often represented by surfaces, where the x and y values contribute to height (z-axis).
• **Approach:** To analyze or solve these functions, approaches such as factoring, expanding, or substituting may be used.
• **Applications:** These functions are vital in fields like economics, physics, and engineering for modeling phenomena that depend on multiple factors, such as pressure and volume in physics or demand and supply in economics.

Understanding how these functions operate helps in evaluating their behavior, especially under varying conditions or limits.

Continuous Functions

Continuous functions in calculus have no breaks, jumps, or holes in their domain. This means you can draw the graph of a continuous function without lifting your pen. Continuity is a key concept because it often simplifies the analysis of functions and their limits.

For the function \(f(x, y) = 4x^2 - 10y^2 + 6\), continuity directly impacts how we assessed the limit.

• **Criteria for Continuity:** A function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point.
• **Continuous in this case:** In our problem, the function is simplified and smooth. When we evaluate it at \((0, 0)\), the function's value \(f(0,0) = 6\) is the same as the limit. Hence, it's continuous at that point.
• **Benefits of Continuity:** If a function is continuous over an interval or domain, we can predict its behavior and easily solve limits involving the function.

Continuous functions, like the one we evaluated, exhibit predictable behavior which greatly assists in problem-solving in calculus.