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.