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When evaluating multivariable limits, it's often helpful to check the behavior of the function along specific pathways. The x-axis is a natural choice because it reduces the complexity by setting one of the variables to zero. In the exercise given, we evaluate the limit as - Set \( y = 0 \). This choice simplifies the expression to focus solely on \( x \), transforming it into a single-variable limit problem: - \( \lim _{x \rightarrow 0} \frac{x \cdot 0^2}{x^2+0^4} = \lim _{x \rightarrow 0} \frac{0}{x^2} = 0 \).Notice that when \( y = 0 \), the numerator becomes zero, leading directly to a limit of zero. This evaluation along the x-axis is crucial as initial insight into the multivariable behavior and simplifies our calculations. However, confirming the limit purely along the x-axis is not enough to conclude the overall limit for multivariable functions.

The y-axis provides another straightforward path to examine the behavior of a multivariable function. It involves simplifying by setting \( x = 0 \), which again turns the multivariable problem into a single-variable task. - Here, by letting \( x = 0 \), the expression simplifies to - \( \lim _{y \rightarrow 0} \frac{0 \cdot y^2}{0^2 + y^4} = \lim _{y \rightarrow 0} \frac{0}{y^4} = 0 \).This approach again results in a zero value, indicating that along the y-axis, the limit tends towards zero. When using these axis limits, you essentially create a kind of test to see if the limit might exist. Zero along both axes is consistent with the existence of a limit, but not conclusive, especially in cases where the function behaves differently off these straightforward paths.

In multivariable calculus, examining the limit along various paths is a common technique to ensure it exists. While axis limits are helpful, they are not sufficient by themselves. For a limit to exist at a point for a multivariable function, it must yield the same result regardless of the path taken to approach that point.- We test various paths such as - \( y = x \) and - \( y = x^2 \)to deepen our understanding.Evaluating along the line \( y = x \), transforms the expression to - \( \lim_{x \rightarrow 0} \frac{x \, x^2}{x^2 + x^4} \).This simplifies further to - \( \lim_{x \rightarrow 0} \frac{x}{1 + x^2} = 0 \).Similarly, along the path \( y = x^2 \), it becomes - \( \lim_{x \rightarrow 0} \frac{x^5}{x^2 + x^8} \), which simplifies to - \( \lim_{x \rightarrow 0} \frac{x^3}{1 + x^6} = 0 \).By performing these additional path tests, we check for path dependency in the function’s behavior as it approaches the origin. If limits differ with different paths, the limit does not exist. Consistent results across these scenarios build confidence about the limit's existence.

Knowing how to evaluate multivariable limits involves using a combination of strategies and insights to fully understand a function's behavior. The process begins with simple paths such as the x and y axes. Here are some techniques to remember:

• **Compute along simple paths:** Start with x-axis and y-axis assessments to look for straightforward results.
• **Try lines and curves:** Evaluate along lines like \( y = x \) or curves like \( y = x^2 \) to see if different functions influence the outcome.
• **Indeterminate case handling:** Understanding how to simplify indeterminate forms, using algebraic manipulation to expose clear limits, is vital.
• **Consistency checks:** Ensure limits are tested with multiple paths to validate the uniformity of results.

By following these techniques, you’ll confirm whether a limit exists. If every tested path yields the same number, it suggests a uniform approach to the origin, consistent with the limit existing. However, differing results across different paths can indicate the lack of a convergent limit.