Problem 73 - 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. $$ \lim _{(x, y) \rightarrow(1,2)}\left(x^{2} y^{3}-x^{3} y^{2}+3 x+2 y\right) $$

Step-by-Step Solution

Answer

The limit is 11.

Step 1: Identify the Function

First, identify the function given in the exercise, which is a multivariable function: \( f(x, y) = x^2 y^3 - x^3 y^2 + 3x + 2y \). We need to find the limit as \((x,y)\) approaches \((1,2)\).

Step 2: Substitute the Values

Substitute \(x = 1\) and \(y = 2\) into the function. This involves replacing every instance of \(x\) and \(y\) in the function with these values.

Step 3: Calculate Each Term Separately

Evaluate each term in the function separately using the substituted values: - The first term: \(x^2 y^3 = 1^2 \, 2^3 = 1 \, 8 = 8\).- The second term: \(x^3 y^2 = 1^3 \, 2^2 = 1 \, 4 = 4\).- The third term: \(3x = 3 \, 1 = 3\).- The fourth term: \(2y = 2 \, 2 = 4\).

Step 4: Combine the Calculated Values

Combine all the calculated values from Step 3:\[8 - 4 + 3 + 4 = 11\]

Step 5: Conclusion

Since all calculations are correct and no zero denominators or undefined expressions appear, the limit as \((x, y)\) approaches \((1, 2)\) exists.

Key Concepts

Limit Evaluation

Evaluating the limit of a multivariable function is fundamental in calculus. Here, the goal is to determine the limit of the function as the independent variables approach certain values. When dealing with multivariable limits, both the path and the point to which the variables converge are important.

• To evaluate the limit, identify the function and the point of interest. In this problem, the point is \( (1,2) \) and the function is \( f(x, y) = x^2 y^3 - x^3 y^2 + 3x + 2y \).
• The existence of the limit depends on whether all paths leading to the point result in the same value.
• Common techniques include direct substitution, which is valid when the function is continuous at the point of interest.

Understanding these basics allows you to apply correct procedures ensuring accurate calculation and interpretation of limits in multivariable calculus.

Substitution Method

The substitution method is a straightforward technique for evaluating limits, particularly useful when dealing with continuous functions. This exercise uses this method effectively for limit determination.

• To use substitution, directly replace the variables in the function with their respective limit values.
• If the function is not continuous at the point, consider checking other methods, because direct substitution might not be the suitable choice.
• Here, the function is smooth and differentiable at \( (1,2) \), which makes substitution especially efficient.

After substitution, evaluate each term to ensure straightforward computation and accuracy in limit evaluation.

Multivariable Calculus

Multivariable calculus expands the ideas of limits, derivatives, and integrals to functions of several variables. This is crucial for analyzing functions in fields such as physics and engineering where more than one independent variable is involved.

• Functions depend on more than one variable, making the computation more complex but also more applicable to real-world problems.
• Understanding the behavior of functions as their variables simultaneously approach certain values requires careful analysis, often employing paths to ensure consistent limit results.
• The function in this problem, involving both \( x \) and \( y \), demonstrates the typical complexity faced in multivariable calculus.

A solid grasp of these principles aids in solving diverse problems involving surfaces and interfaces in higher dimensions.

Function Analysis

Analyzing a function carefully often involves a break-down of its components, facilitating easier understanding and evaluation of expressions. In multivariable calculus, this analysis includes:

• Breaking the function into simpler terms, assessing each one independently, as seen where \( x^2 y^3 \) and \( x^3 y^2 \) are separately calculated.
• Exploring the interaction between terms, which can look different from single-variable calculus due to multiple variables mixing in each term.
• Observing how changes in input values reflect through each component of the function to yield the final result.

This stepwise approach not only simplifies computations but also enhances comprehension, crucial for tackling more advanced calculus problems with confidence.