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The image presents a mathematical problem asking to find the limit of the function \(\frac{(-7x + y)^2}{49x^2 + y^2}\) as \((x, y) \to (0,0)\) along different paths, and then determine whether the limit exists.

The paths to be analyzed are:
1. Along the x-axis (where \( y = 0 \))
2. Along the y-axis (where \( x = 0 \))
3. Along the line \( y = x \))

Let's calculate the limit for each path to verify the answers provided and then determine whether the overall limit exists.

1. **Along the x-axis:** Here, \( y = 0 \).
   \[
   \lim_{(x,0) \to (0,0)} \frac{(-7x + 0)^2}{49x^2 + 0^2} = \lim_{x \to 0} \frac{49x^2}{49x^2} = 1
   \]

2. **Along the y-axis:** Here, \( x = 0 \).
   \[
   \lim_{(0,y) \to (0,0)} \frac{(-7 \cdot 0 + y)^2}{49 \cdot 0^2 + y^2} = \lim_{y \to 0} \frac{y^2}{y^2} = 1
   \]

3. **Along the line \( y = x \):**
   \[
   \lim_{(x,x) \to (0,0)} \frac{(-7x + x)^2}{49x^2 + x^2} = \lim_{x \to 0} \frac{(-6x)^2}{50x^2} = \lim_{x \to 0} \frac{36x^2}{50x^2} = \frac{36}{50} = 0.72
   \]

From this calculation, it appears that there is an error in the provided answer for the limit along the line \( y = x \). The actual value should be \(0.72\) and not \(0\) as stated.

4. **Existence of the limit:**
   Since the limit values differ along different paths (1 along the x-axis and y-axis, but 0.72 along \( y = x \)), the overall limit of the function as \((x, y) \to (0,0)\) does not exist.

These calculations show that the given answer for the overall limit ("DNE" or does not exist) is correct. However, the specific value along \( y = x \) should be \(0.72\), not \(0\).

Would you like to delve deeper into any specific part of this problem or have any other math-related questions?

**Additional Queries You Might Consider:**
1. How to find limits of functions of two variables using polar coordinates.
2. How does one generally prove that a limit does not exist?
3. The importance of paths in determining the limits in multivariable calculus.
4. Examples of functions with path-dependent limits.
5. How could changing the function slightly affect the limit along different paths?

**Tip:** Always verify the path-specific results when evaluating the overall limit in multivariable functions, as this helps in understanding the behavior of the function near the point of interest.

Find the limit of ( (-7x + y)^2 ) / (49x^2 + y^2) as (x, y) approaches (0, 0) along different paths.

Determine the limit of the function ((-7x + y)^2)/(49x^2 + y^2) as (x, y) approaches (0,0) along different paths such as the x-axis, y-axis, and the line y = x. Understand path-dependent limits in multivariable calculus.

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