Problem 15
Question
Evaluate the limit along the paths given, then state why these results show the given limit does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) (a) Along the path \(y=0\). (b) Along the path \(x=0\).
Step-by-Step Solution
Verified Answer
The limit does not exist because different paths give different results (along y=0: 1, along x=0: -1).
1Step 1: Substitute Along y=0
Substitute the path where \( y = 0 \) into the function. This gives: \[ f(x, y) = \frac{x^2 - 0^2}{x^2 + 0^2} = \frac{x^2}{x^2} = 1 \] Thus, the limit along this path as \( (x, y) \rightarrow (0,0) \) is 1.
2Step 2: Substitute Along x=0
Substitute the path where \( x = 0 \) into the function. This gives: \[ f(x, y) = \frac{0^2 - y^2}{0^2 + y^2} = \frac{-y^2}{y^2} = -1 \] Thus, the limit along this path as \( (x, y) \rightarrow (0,0) \) is -1.
3Step 3: Compare Limits on Different Paths
The limits we obtained by evaluating along the two different paths are different: 1 and -1. In multivariable calculus, if the limit along two different paths is not the same, the overall limit does not exist. Thus, \( \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \) does not exist.
Key Concepts
Limit EvaluationMultivariable CalculusPath-Dependent Limits
Limit Evaluation
In single-variable calculus, evaluating limits involves approaching a point from two directions: the left and the right. However, multivariable calculus adds more complexity since points can be approached from an infinite number of paths in a plane or space. Evaluating limits in multivariable calculus is about finding out if the value of a function approaches a single unique number no matter which way you approach a specific point.
If you find that approaching the point via different paths results in different limit values, it indicates that the overall limit does not exist. This is because limits in multiple dimensions must be path-independent. Any difference in limit values along distinct paths reveals that no single number is consistently approached. Thus, checking multiple paths is key to determining the existence of a limit in multivariable settings.
If you find that approaching the point via different paths results in different limit values, it indicates that the overall limit does not exist. This is because limits in multiple dimensions must be path-independent. Any difference in limit values along distinct paths reveals that no single number is consistently approached. Thus, checking multiple paths is key to determining the existence of a limit in multivariable settings.
Multivariable Calculus
Multivariable calculus extends the principles of calculus to functions with more than one variable, typically represented as functions of the form \( f(x, y) \) or \( f(x, y, z) \). This branch of calculus is crucial for understanding phenomena where multiple inputs affect the outcome, common in fields like physics, engineering, and economics.
- Functions: Deals with understanding and analyzing functions with two or more variables.
- Partial Derivatives: Involves differentiating functions with respect to one variable, holding others constant.
- Gradients: Gives the direction and rate of steepest ascent in a function.
- Multiple Integrals: Allows calculation of area and volume in higher dimensions.
Path-Dependent Limits
Path-dependent limits occur when a function does not settle on a single value as it approaches a point from different trajectories within its domain. Such behavior is specific to multivariable calculus, where paths in a plane or space can vary greatly.
To understand path-dependent limits, consider that as you approach a point \((x,y)\), you might do so along a straight line like \(y = 0\) or \(x = 0\), or even along curves like \(y = x^2\). If each path gives a different limit, then the function's limit at that point is path-dependent and does not exist in the traditional sense.
In the provided exercise, evaluating the limit along the lines \(y = 0\) and \(x = 0\) resulted in different values (1 and -1, respectively). Since these results differ, the limit is path-dependent, demonstrating that no single value can be defined for all paths approaching \((0,0)\). Thus, the overall limit does not exist, offering a clear demonstration of how path-dependent limits affect calculations in multivariable calculus.
To understand path-dependent limits, consider that as you approach a point \((x,y)\), you might do so along a straight line like \(y = 0\) or \(x = 0\), or even along curves like \(y = x^2\). If each path gives a different limit, then the function's limit at that point is path-dependent and does not exist in the traditional sense.
In the provided exercise, evaluating the limit along the lines \(y = 0\) and \(x = 0\) resulted in different values (1 and -1, respectively). Since these results differ, the limit is path-dependent, demonstrating that no single value can be defined for all paths approaching \((0,0)\). Thus, the overall limit does not exist, offering a clear demonstration of how path-dependent limits affect calculations in multivariable calculus.
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