Problem 10 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The value of the limit is -1.

Step 1: Understand the Expression

We want to find the limit of the expression \(xy^2z^3\) as \((x, y, z)\) approaches \((1, 1, -1)\). In other words, we want to find the value of the expression as \(x \to 1\), \(y \to 1\), and \(z \to -1\).

Step 2: Substitute the Limiting Values

As \((x, y, z)\) approaches \((1, 1, -1)\), we substitute the values \(x=1\), \(y=1\), and \(z=-1\) into the expression \(xy^2z^3\): $$\lim_{(x, y, z) \rightarrow (1, 1, -1)} (1)(1^2)(-1)^3$$

Step 3: Compute the Limit

Simplify the expression in the previous step: $$(1)(1^2)((-1)^3) = (1)(1)(-1) = -1$$

Step 4: Write the Final Answer

Therefore, the value of the limit as \((x, y, z)\) approaches \((1, 1, -1)\) for the expression \(xy^2z^3\) is: $$\lim_{(x, y, z) \to (1, 1, -1)}xy^2z^3 = -1$$

Key Concepts

Limits in Three Variables

In multivariable calculus, limits extend beyond just the x-variable to include more dimensions like y and z, helping us study the behavior of functions in a multi-dimensional space. This concept allows us to understand how functions behave as inputs approach a particular point
across three dimensions.
For the problem given, we evaluate the limit of the expression as \((x, y, z)\) approaches \((1, 1, -1)\). The goal is to determine the value of the expression \(xy^2z^3\) when \(x\), \(y\), and \(z\) are close to \(1\), \(1\), and \(-1\) respectively.

• This involves identifying the three-dimensional point, which in this case is \((1, 1, -1)\).
• We then evaluate how the expression behaves as each variable approaches its respective limiting value simultaneously.

Understanding this multi-directional approach to limits is crucial, especially in functions that include multiple variables. As each variable approaches its target value, the collective behavior can show complexity and requires special computation methods to find a precise answer.

Substitution Method in Calculus

One simple technique for evaluating limits in calculus is the substitution method. This method involves directly substituting the approaching values of the variables into the expression. If the function and approaching values are straightforward, substitution provides a direct path to the limit.When using the substitution method, follow these clear steps:

• Identify the limiting values given by the multi-variable point. For this problem, those values are \(x = 1\), \(y = 1\), and \(z = -1\).
• Substitute these values into the expression, \(xy^2z^3\), to find \((1)(1^2)(-1)^3\).
• Calculate the expression post-substitution to simplify how it behaves near the point of interest.

This method is a quick and effective way to resolve limits for uncomplicated expressions without indeterminate forms. However, caution should be exercised since not all expressions can be directly substituted if they result in undefined forms.

Limits Evaluation Steps

Evaluating limits, especially in multivariable calculus, involves a series of logical steps that ensure clarity and accuracy. The process is largely systematic. Let's outline them based on the problem given:1. **Understand the Expression:** Begin by clearly defining the expression whose limit you need to evaluate. In this instance, it is \(xy^2z^3\).
2. **Identify the Point of Approach:** The specific point \((x, y, z)\) approaches is critical. Here, it's \((1, 1, -1)\), the point where \(x\) becomes \(1\), \(y\) approaches \(1\), and \(z\) tends towards \(-1\).
3. **Substitute Limiting Values:** Substitute these values into the expression. For \(xy^2z^3\), replace \(x\) with \(1\), \(y\) with \(1\), and \(z\) with \(-1\).
4. **Compute the Limit:** Perform the necessary calculations. Instantiate \((1)(1^2)(-1)^3\) and simplify it to get \(-1\).
5. **Write Down the Result:** Express the final answer clearly: \(\lim_{(x, y, z) \to (1, 1, -1)}xy^2z^3 = -1\).
This structured approach ensures precision and allows you to consistently find the correct limit values for expressions involving multiple variables.