Problem 5 - Calculus Early Transcendentals

Question

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(1,2)}\left(5 x^{3}-x^{2} y^{2}\right)$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Problem

We are tasked with finding the limit of the function \(f(x, y) = 5x^3 - x^2y^2\) as \((x, y)\) approaches \((1, 2)\). This involves checking if the limit exists as the point approaches from different paths.

Step 2: Substituting Directly into the Function

Start by substituting \(x = 1\) and \(y = 2\) directly into the function, \(f(x, y)\):\[ f(1, 2) = 5(1)^3 - (1)^2(2)^2 = 5 \times 1 - 1 \times 4 = 5 - 4 = 1 \]

Step 3: Approaching Along the X-Axis

Consider the path where \(y = 2\). Substitute \(y = 2\) into \(f(x, y)\), we get:\[ g(x) = 5x^3 - x^2(2)^2 = 5x^3 - 4x^2 \]Taking the limit as \(x \to 1\) gives:\[ \lim_{x \to 1} (5x^3 - 4x^2) = 5(1)^3 - 4(1)^2 = 5 - 4 = 1 \]

Step 4: Approaching Along the Y-Axis

Consider the path where \(x = 1\). Substitute \(x = 1\) into \(f(x, y)\), we get:\[ h(y) = 5(1)^3 - (1)^2y^2 = 5 - y^2 \]Taking the limit as \(y \to 2\) gives:\[ \lim_{y \to 2} (5 - y^2) = 5 - (2)^2 = 5 - 4 = 1 \]

Step 5: Conclusion: Limit Exists

Since the limit along both paths (and direct substitution) give the same result of 1, the directional limits are consistent. Thus, the limit exists and is equal to 1.

Key Concepts

Limits in Multivariable Functions

Understanding limits in multivariable functions is a step beyond single-variable calculus. In multivariable calculus, we deal with functions that depend on more than one variable, say, variables like \(x\) and \(y\). A limit in this context describes the approach of function values to a particular point or number as the variables \((x, y)\) approach a certain point \((a, b)\). The limit tells us how the function behaves, or tends to behave, closer to the target point. \( \lim_{(x, y) \to (a, b)} f(x, y) \) Measures how \(f(x, y)\) behaves as \(x\) approaches \(a\) and \(y\) approaches \(b\). Calculating this limit can involve substituting the values of \((a, b)\) directly into the equation and seeing if the function outputs a finite number. If it results in a finite value for all possible paths towards \((a, b)\), the limit exists.

Directional Limits

Directional limits in multivariable calculus address the challenge arising from having multiple paths to approach a point. Unlike single-variable functions, multivariable functions allow us to approach a point from different directions, not just from the left or right. Each direction can potentially yield a different limit. To confirm that a limit for a multivariable function exists, we need to ensure that it is the same from all directions.
For example, when calculating the limit of \(f(x, y)\) as \((x, y) \to (1, 2)\), we can approach along various paths, such as the x-axis, y-axis, or even diagonals.

• Approaching directly: Substitute \(x = 1\) and \(y = 2\).
• A long the x-direction: Hold \(y=2\) and let \(x\) vary to 1.
• A long the y-direction: Hold \(x=1\) and let \(y\) vary to 2.

All these paths should lead to the same result for the limit to exist at that point.

Path Independence of Limits

Path independence is crucial when exploring limits in functions of several variables. It asserts that for a limit to exist in multivariable calculus, it must not change regardless of the chosen path. This means whether you approach \((a, b)\) directly, along axes, or through other paths like curves such as \(y = mx + b\), the result must be the same.
In the given exercise, after evaluating different paths to approach the point \((1, 2)\), if all paths lead to the same value 1, it confirms that the limit is path-independent and truly exists.

• Using direct substitution.
• Approaching along axes.
• Trying unusual curves or paths between the points.

All should consistently provide the value of 1 to assert that the limit is definitively 1, regardless of path.

Calculus Techniques

To successfully solve limits in multivariable calculus, deploying proper calculus techniques is essential.
In the step-by-step approach to solving limits like the one in the exercise provided, these techniques are used:

• Direct Substitution: Often the first step, it simplifies the process by substituting the target values directly into the function. If this yields a finite number, it's a strong indicator of the limit at that point.
• Path Consideration: Knowing that different paths might provide different values, trying at least two paths help ensure result consistency. This includes simple straight-line paths like axes and through curves.
• Simplification: Simplifying expressions when stuck with complex algebra can make evaluation easier and faster.

Using these foundational techniques, particularly for checking multiple paths and simplifying expressions, provides deeper insights into the behavior of multivariable functions as they approach particular points.