Problem 61 - Calculus 3

Question

For the following exercises, find the limit of the function. $$ \lim _{(x, y) \rightarrow(1,2) x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

The limit is 5.

Step 1: Understand the Limit

Here we need to find the limit of the function as \((x, y)\) approaches \((1, 2)\). The given function is \(x^2 + y^2\). This means we substitute \(x = 1\) and \(y = 2\) into the function to evaluate the limit.

Step 2: Substitute the Values

Substitute \(x = 1\) and \(y = 2\) directly into the expression \(x^2 + y^2\). So, the expression becomes \(1^2 + 2^2\).

Step 3: Calculate the Expression

Now calculate the value of the expression obtained in Step 2. Compute \(1^2 = 1\) and \(2^2 = 4\), then add them: \(1 + 4 = 5\).

Step 4: Conclude the Limit

Since the variable substitution is direct, and as both \(x\) and \(y\) approach \( (1, 2) \), we have found the limit of the function \(x^2 + y^2\) at \((1,2)\) to be \(5\).

Key Concepts

Understanding Multivariable Functions

Multivariable functions are mathematical expressions involving more than one variable. Unlike single-variable functions, which have the form of \(f(x)\), multivariable functions involve at least two inputs, commonly written as \(f(x, y)\). These functions map pairs or sets of numbers to a single output
. Understanding multivariable functions is crucial since they can represent real-world phenomena, such as surfaces or three-dimensional shapes.Consider the function \(f(x, y) = x^2 + y^2\). Here, each ordered pair \((x, y)\) gives a corresponding output value determined by squaring \(x\), squaring \(y\), and adding the two results. This particular function can be visualized as a parabolic surface that opens upwards.Some key points about multivariable functions:

• They involve more than one variable.
• Each variable can affect the output in different ways.
• These functions are often used in calculus to analyze surfaces or regions in a multivariable space.

When dealing with such functions, visualization and a good understanding of geometric interpretations often prove beneficial.

The Process of Function Evaluation

Function evaluation is the process of determining the output value of a function for given input values. For multivariable functions, this often involves substituting specified values for each variable.
For instance, in the function \(x^2 + y^2\), you may need to evaluate the function at \((1, 2)\) to find out what the function equals when \(x = 1\) and \(y = 2\). To do this, replace \(x\) with \(1\) and \(y\) with \(2\) in the expression, yielding \(1^2 + 2^2\). The calculation process is straightforward:

• Square each substituted value: \(1^2\) and \(2^2\).
• Add the squared values: \(1 + 4 = 5\).

The final result, \(5\), represents the value of the function at that point. It highlights the importance of correct substitution and calculation to evaluate functions accurately.
Function evaluation is a basic yet essential skill in calculus, enabling students to handle more complex expressions and analyze diverse mathematical models effectively.

Limit Calculation in Multivariable Contexts

Limit calculation involves finding what a function approaches as the inputs get close to a certain point. In multivariable calculus, this means looking at the behavior of functions as the point \((x, y)\) approaches \((a, b)\). Calculating limits in such a setting requires thorough understanding, as small errors might lead to incorrect results.
Take for example the expression \(\lim_{(x, y) \rightarrow (1, 2)} x^2 + y^2\). To solve this limit, you evaluate the function \(x^2 + y^2\) by substituting \(x = 1\) and \(y = 2\). Here's the process:

• Substitute: Replace \(x\) with \(1\) and \(y\) with \(2\) in \(x^2 + y^2\).
• Evaluate: Compute the expression: \(1^2 + 2^2 = 5\).

The limit equals \(5\), indicating that as \((x, y)\) gets very close to \((1, 2)\), the function approaches \(5\). This concept can be extended to analyze more complex expressions or scenarios where intuitive computational methods aren't immediately apparent. Handling limits effectively requires understanding paths of approach and sometimes employing techniques such as polar coordinates or using software tools to verify computations.