find the limit of $$ g(x, y) $$ as $$ (x, y) $$ - $$ \left(x_{0}, y_{0}\right) $$ aling a shaight line pat if $$ g(x, y)=\frac{2 x-y}{x^{2}+y^{2}},\left(x_{0}, y_{0}\right)=(0,1) $$

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To find the limit of the function $$ g(x, y) = \frac{2x - y}{x^2 + y^2} $$ as $$(x, y)$$ approaches $$(0, 1)$$ along a straight line path, we'll consider a general straight line approaching $$(0, 1)$$.

### Step 1: Parameterize the Straight Line Path

Let’s parameterize the path as:
$$
y = 1 + m x
$$
where $$ m $$ is the slope of the line. As $$ x $$ approaches 0, $$ y $$ approaches 1.

### Step 2: Substitute into $$ g(x, y) $$

Substitute $$ y = 1 + m x $$ into the function:
$$
g(x, y) = \frac{2x - (1 + m x)}{x^2 + (1 + m x)^2}
$$
Simplify the numerator and the denominator:
$$
\text{Numerator} = 2x - 1 - m x = (2 - m)x - 1
$$
$$
\text{Denominator} = x^2 + 1 + 2m x + m^2 x^2 = 1 + 2m x + (1 + m^2)x^2
$$

### Step 3: Evaluate the Limit as $$ x \to 0 $$

As $$ x $$ approaches 0:
$$
\text{Numerator} \to -1
$$
$$
\text{Denominator} \to 1
$$
Thus,
$$
\lim_{(x, y) \to (0, 1)} g(x, y) = \frac{-1}{1} = -1
$$

### Conclusion

Regardless of the slope $$ m $$ of the straight line path chosen, the limit of $$ g(x, y) $$ as $$(x, y)$$ approaches $$(0, 1)$$ along that path is:

$$\boxed{-1}$$
The limit of $$ g(x, y) = \frac{2x - y}{x^2 + y^2} $$ as $$(x, y)$$ approaches $$(0, 1)$$ along any straight line path is $$-1$$.

find the limit of as -
aling a shaight line pat
if

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find the limit of as - aling a shaight line pat if