Problem 4
Question
Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1,-1)}\left(2 x^{3}-3 y\right)(x y+1)\)
Step-by-Step Solution
Verified Answer
The limit is 0 as \((x, y)\) approaches \((1, -1)\).
1Step 1: Identify the Components
The given expression is \((2x^3 - 3y)(xy + 1)\). It involves two expressions being multiplied together: \(2x^3 - 3y\) and \(xy + 1\). We need to find their limits as \(x\) approaches 1 and \(y\) approaches -1.
2Step 2: Calculate the Limit of the First Expression
First, we will find the limit of \(2x^3 - 3y\) as \((x, y)\) approaches \((1, -1)\). Substitute \(x = 1\) and \(y = -1\): \[2(1)^3 - 3(-1) = 2 + 3 = 5.\] So, the limit of the first expression is 5.
3Step 3: Calculate the Limit of the Second Expression
Next, we calculate the limit of \(xy + 1\) as \((x, y)\) approaches \((1, -1)\). Substitute \(x = 1\) and \(y = -1\): \[(1)(-1) + 1 = -1 + 1 = 0.\] Thus, the limit of the second expression is 0.
4Step 4: Apply the Multiplication Limit Rule
We now apply the rule that states the limit of a product is the product of the limits (if each limit exists). Therefore, the limit of the expression \((2x^3 - 3y)(xy + 1)\) is: \[5 \times 0 = 0.\] Thus, the overall limit of the given expression is 0 as \((x, y)\) approaches \((1, -1)\).
Key Concepts
Multivariable CalculusLimit PropertiesProduct Rule for Limits
Multivariable Calculus
Multivariable calculus is an extension of single-variable calculus to functions of more than one variable. In this exercise, we deal with a function in two variables: \(x\) and \(y\). Understanding how limits work in this context is crucial since it allows us to analyze functions in a multidimensional space.
In multivariable calculus, the limit of a function \(f(x, y)\) as \((x, y)\) approaches a point \((a, b)\) examines the behavior of \(f\) as its variables get arbitrarily close to \(a\) and \(b\). This is analogous to how limits work with single-variable functions but requires considering paths of approach from multiple directions.
It's important to remember when evaluating limits in multivariable calculus that just like in single-variable functions, the limit must be the same regardless of the approach path. If a different path gives different limits, the limit does not exist.
In multivariable calculus, the limit of a function \(f(x, y)\) as \((x, y)\) approaches a point \((a, b)\) examines the behavior of \(f\) as its variables get arbitrarily close to \(a\) and \(b\). This is analogous to how limits work with single-variable functions but requires considering paths of approach from multiple directions.
It's important to remember when evaluating limits in multivariable calculus that just like in single-variable functions, the limit must be the same regardless of the approach path. If a different path gives different limits, the limit does not exist.
Limit Properties
Understanding limit properties simplifies evaluating complex expressions. Properties like the sum, difference, scalar multiplication, and the product of limits often facilitate calculations. These properties can be especially beneficial when dealing with multivariable functions.
Let's focus on two properties used in our solution:
Let's focus on two properties used in our solution:
- Limit of a Sum: The limit of a sum of two functions equals the sum of the limits of the individual functions, if they exist. Mathematically, if \(\lim_{(x, y) \to (a, b)} f(x, y) = L\) and \(\lim_{(x, y) \to (a, b)} g(x, y) = M\), then \(\lim_{(x, y) \to (a, b)} [f(x, y) + g(x, y)] = L + M\).
- Limit of a Difference: Similar to the limit of a sum, the limit of a difference is the difference of the individual limits.
Product Rule for Limits
The product rule for limits is an instrumental principle in calculus. It states that the limit of a product of two functions is the product of their limits, provided that each of these limits exists. In mathematical terms, if \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), then \(\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M\).
In the exercise at hand, we applied this rule to two expressions, \(2x^3 - 3y\) and \(xy + 1\). Once the individual limits (5 and 0, respectively) were found, we multiplied them to find the overall limit.
The product rule simplifies the often daunting task of handling a function that results from combining two separate functions. It's essential to check the existence of both individual limits to apply this rule correctly. If either limit does not exist, the limit of the product cannot be determined using this rule.
In the exercise at hand, we applied this rule to two expressions, \(2x^3 - 3y\) and \(xy + 1\). Once the individual limits (5 and 0, respectively) were found, we multiplied them to find the overall limit.
The product rule simplifies the often daunting task of handling a function that results from combining two separate functions. It's essential to check the existence of both individual limits to apply this rule correctly. If either limit does not exist, the limit of the product cannot be determined using this rule.
Other exercises in this chapter
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