Problem 1
Question
In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(1,0)}\left(x^{2}-3 y^{2}\right) $$
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: State the Problem
We need to calculate the limit of the expression \(x^2 - 3y^2\) as \((x, y)\) approaches \((1, 0)\).
2Step 2: Apply Limit Properties
The limit of a sum or difference is the sum or difference of the limits. Thus, we can separate the two terms and compute: \[ \lim_{(x, y) \to (1, 0)} (x^2 - 3y^2) = \lim_{(x, y) \to (1, 0)} x^2 - \lim_{(x, y) \to (1, 0)} 3y^2 \]
3Step 3: Compute Each Limit Separately
Compute \(\lim_{(x, y) \to (1, 0)} x^2\). Here, as \(x\) approaches 1, \(x^2\) approaches \(1^2\), which is 1. Now compute \(\lim_{(x, y) \to (1, 0)} 3y^2\). As \(y\) approaches 0, \(3y^2\) approaches \(3 \times 0^2 = 0\).
4Step 4: Combine the Results
Substitute back the limits found: \(\lim_{(x, y) \to (1, 0)} x^2 - 3y^2 = 1 - 0 = 1\). Thus, the limit is 1.
Key Concepts
Limit PropertiesMultivariable CalculusLimit Evaluation
Limit Properties
When studying limits, it's essential to understand how they can be managed using mathematical properties.
Limit properties act like tools that make limit evaluation more efficient and approachable.
Here are some key properties you'll want to remember when working with limits:
Limit properties act like tools that make limit evaluation more efficient and approachable.
Here are some key properties you'll want to remember when working with limits:
- The limit of a sum is the sum of the limits: \[\lim_{{x \to a}} (f(x) + g(x)) = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)\]
- The limit of a difference is the difference of the limits: \[\lim_{{x \to a}} (f(x) - g(x)) = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)\]
- The limit of a product is the product of the limits: \[\lim_{{x \to a}} (f(x) \cdot g(x)) = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)\]
- The limit of a quotient is the quotient of the limits (provided the denominator is not zero): \[\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)}\]
Multivariable Calculus
Multivariable calculus, unlike single-variable calculus, deals with functions that have more than one independent variable.
This branch of calculus helps us explore more complex systems and scenarios met in real-world applications.
In multivariable calculus, when we evaluate limits, we're looking at how a function behaves as we change two or more variables simultaneously.
For example, in the expression \[f(x, y) = x^2 - 3y^2\], we have two variables, \(x\) and \(y\).
As a result, thinking about the limit becomes a bit more involved because we examine how the result changes as both \(x\) and \(y\) head towards specific values.
For the given example, we investigated how the function changes as \[(x, y) \to (1, 0)\].
That means both \(x\) approaches 1 and \(y\) approaches 0 at the same time, a fundamental principle to grasp when entering the world of multivariable calculus.
Understanding paths when limits of multivariable functions are involved is key, as different paths can sometimes yield different results.
This branch of calculus helps us explore more complex systems and scenarios met in real-world applications.
In multivariable calculus, when we evaluate limits, we're looking at how a function behaves as we change two or more variables simultaneously.
For example, in the expression \[f(x, y) = x^2 - 3y^2\], we have two variables, \(x\) and \(y\).
As a result, thinking about the limit becomes a bit more involved because we examine how the result changes as both \(x\) and \(y\) head towards specific values.
For the given example, we investigated how the function changes as \[(x, y) \to (1, 0)\].
That means both \(x\) approaches 1 and \(y\) approaches 0 at the same time, a fundamental principle to grasp when entering the world of multivariable calculus.
Understanding paths when limits of multivariable functions are involved is key, as different paths can sometimes yield different results.
Limit Evaluation
Evaluating limits in multivariable calculus is a crucial skill and sometimes involves more steps than single-variable calculus.
In our example, we evaluated the expression \[\lim_{(x, y) \to (1, 0)} (x^2 - 3y^2)\].
Here's a step-by-step approach showcasing how we evaluate such limits using limit properties.
Similarly, for \(\lim_{(x, y) \to (1, 0)} 3y^2\), substituting \(y = 0\) results in a value of 0.
Thus, the limit of this function as \((x, y) \to (1, 0)\) is 1.
This systematic break-down simplifies complex problems into manageable parts, making limit evaluation clear and straightforward.
In our example, we evaluated the expression \[\lim_{(x, y) \to (1, 0)} (x^2 - 3y^2)\].
Here's a step-by-step approach showcasing how we evaluate such limits using limit properties.
Breaking Down the Problem
To start, we break the expression \(x^2 - 3y^2\) into two separate limits using the property of differences of limits.Compute Each Part Separately
Next, you compute each part separately. This involves considering \(\lim_{(x, y) \to (1, 0)} x^2\), where we substitute \(x = 1\), and this equals 1.Similarly, for \(\lim_{(x, y) \to (1, 0)} 3y^2\), substituting \(y = 0\) results in a value of 0.
Combining Results
Finally, combine the results using the difference of these computed limits to find:\( 1 - 0 = 1\).Thus, the limit of this function as \((x, y) \to (1, 0)\) is 1.
This systematic break-down simplifies complex problems into manageable parts, making limit evaluation clear and straightforward.
Other exercises in this chapter
Problem 1
In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=x^{2} y+x y^{2} $$
View solution Problem 1
In Problems \(1-10\), the functions are defined for all \((x, y) \in \boldsymbol{R}^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to
View solution Problem 2
Let \(f(x, y)=e^{x} \sin y\) with \(x(t)=t\) and \(y(t)=t^{3}\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=1\).
View solution Problem 2
Mean arterial blood pressure (MAP) is a function of systolic blood pressure (SP) and diastolic blood pressure (DP). At a resting heart rate, $$ \mathrm{MAP} \ap
View solution