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Chapter 12: Problem 10

$$\text { Evaluate } \lim _{(x, y, z) \rightarrow(1,1,-1)} x y^{2} z^{3}$$

Short Answer

Expert verified
Question: Evaluate the limit of the function $$f(x, y, z) = x y^{2} z^{3}$$ as \((x, y, z)\) approaches \((1, 1, -1)\). Answer: The limit is $$\lim _{(x, y, z) \rightarrow(1,1,-1)} x y^{2} z^{3} = -1$$.

Step by step solution

01

Identify the function and the point

The given function is $$f(x, y, z) = x y^{2} z^{3}$$ We need to evaluate the limit of this function as \((x, y, z)\) approaches \((1, 1, -1)\).
02

Substitute the values

Since the function is a polynomial, we can directly substitute the values \((x, y, z) = (1, 1, -1)\) in the function: $$f(1, 1, -1) = 1(1)^2(-1)^3$$
03

Calculate the limit

Now, we can calculate the limit by simplifying the expression obtained in Step 2: $$f(1, 1, -1) = 1(1)(-1)$$ $$f(1, 1, -1) = -1$$ So, the limit is: $$\lim _{(x, y, z) \rightarrow(1,1,-1)} x y^{2} z^{3} = -1$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits in Multiple Dimensions
A fundamental concept in multivariable calculus is understanding how limits work when we deal with functions of more than one variable. Unlike single-variable calculus, here we look at functions as points in a multi-dimensional space. Imagine trying to approach a point in a three-dimensional space from all possible directions. This is what we do with limits in multiple dimensions.
  • How it Works: We consider the limit of a function as the variables approach a particular point from every direction.
  • Function Consideration: The considered function is \(f(x, y, z) = x y^2 z^3\), which takes three inputs, making it a function in three-dimensional space.
  • Approaching a Point: In this specific case, we want to find out what \(f(x, y, z)\) approaches as \(x, y, z\) get closer to \(1, 1, -1\).
To ensure that a limit exists, the function must approach the same value from all paths towards the point. In polynomial functions, this often involves simplifications and substitution, which we'll cover later.
Polynomial Functions in Calculus
Polynomial functions are a core part of calculus and can simplify many problems, especially when dealing with limits. A polynomial function is composed of variables raised to whole-number exponents, multiplied by coefficients, and summed together.
  • Simple Polynomials: These are expressions like \(x + y^2 + z^3\).
  • Our Context: The function \(x y^2 z^3\) is straightforward, with each variable raised to a specific power and multiplied, representing a three-dimensional polynomial function.
Because polynomial functions are continuous everywhere in their domain, taking limits at any point involves direct substitution. Substituting the values gives you the result directly, without needing to consider the paths鈥攚hich simplifies finding limits in multi-dimensional calculus problems.
Substitution Method in Limits
One of the most practical techniques in evaluating limits, especially in polynomial functions, is the substitution method. This simple but powerful tool works well in functions that are continuous at the point of interest.
  • Why Substitution Works: Continous functions allow you to directly substitute the values of the variables to evaluate the limit, provided the point is within the function's domain.
  • Our Example: For the function \(x y^2 z^3\), as \(x, y, z\) approach \(1, 1, -1\), simply rooting the values into the function allows the limit to be calculated quickly: \(1 \cdot 1^2 \cdot (-1)^3 = -1\).
This method is efficient because it avoids needing additional calculations or considerations of directionality, which could complicate the process in non-polynomial functions. It's a crucial concept for students to grasp, enabling them to solve many kinds of limit problems swiftly.

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