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Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Please show steps to find the limit
### Problem: Evaluate the Limit

**Question:**

Find the limit of the following function as \( (x, y) \) approaches \( (0,0) \):

\[
\lim_{(x, y) \to (0,0)} \frac{x^2 y}{x^4 + 4y^2}
\]

**Explanation:**

This mathematical limit problem involves functions of two variables. The function provided is:

\[
\frac{x^2 y}{x^4 + 4y^2}
\]

To solve for the limit as \((x, y)\) approaches \((0,0)\), different methods can be applied, such as substitution of specific paths or recognizing the behavior of the function as both variables tend to zero.

In this case, consider paths along which \(y=kx^2\):

\[
\frac{x^2 (kx^2)}{x^4 + 4(kx^2)^2} = \frac{kx^4}{x^4 + 4k^2x^4} = \frac{kx^4}{x^4 (1 + 4k^2)} = \frac{k}{1 + 4k^2}
\]

When \(x \neq 0\), this simplifies to a constant depending on \(k\). As \(x \to 0\), regardless of \(k\), the expression tends to 0. Therefore, the limit can be evaluated as:

\[
\lim_{(x, y) \to (0,0)} \frac{x^2 y}{x^4 + 4y^2} = 0
\]

Thus, the final answer is:

\[
\boxed{0}
\]
Transcribed Image Text:### Problem: Evaluate the Limit **Question:** Find the limit of the following function as \( (x, y) \) approaches \( (0,0) \): \[ \lim_{(x, y) \to (0,0)} \frac{x^2 y}{x^4 + 4y^2} \] **Explanation:** This mathematical limit problem involves functions of two variables. The function provided is: \[ \frac{x^2 y}{x^4 + 4y^2} \] To solve for the limit as \((x, y)\) approaches \((0,0)\), different methods can be applied, such as substitution of specific paths or recognizing the behavior of the function as both variables tend to zero. In this case, consider paths along which \(y=kx^2\): \[ \frac{x^2 (kx^2)}{x^4 + 4(kx^2)^2} = \frac{kx^4}{x^4 + 4k^2x^4} = \frac{kx^4}{x^4 (1 + 4k^2)} = \frac{k}{1 + 4k^2} \] When \(x \neq 0\), this simplifies to a constant depending on \(k\). As \(x \to 0\), regardless of \(k\), the expression tends to 0. Therefore, the limit can be evaluated as: \[ \lim_{(x, y) \to (0,0)} \frac{x^2 y}{x^4 + 4y^2} = 0 \] Thus, the final answer is: \[ \boxed{0} \]
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