The given problem is to evaluate the following limit:

$\lim_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2 + y^2}}$

Solution:

To determine if the limit exists, we need to check the behavior of the function along different paths approaching the origin $(0, 0)$.

1. Path 1: Along the x-axis (y = 0):

   • Substituting $y = 0$ into the expression: $f(x, 0) = \frac{x(0)}{\sqrt{x^2 + 0^2}} = 0$
   • So, along the x-axis, the limit is $0$.

2. Path 2: Along the y-axis (x = 0):

   • Substituting $x = 0$ into the expression: $f(0, y) = \frac{0(y)}{\sqrt{0^2 + y^2}} = 0$
   • So, along the y-axis, the limit is also $0$.

3. Path 3: Along the line $y = x$:

   • Substituting $y = x$ into the expression: $f(x, x) = \frac{x(x)}{\sqrt{x^2 + x^2}} = \frac{x^2}{\sqrt{2x^2}} = \frac{x^2}{\sqrt{2}|x|} = \frac{|x|}{\sqrt{2}}$
   • As $x \to 0$, $\frac{|x|}{\sqrt{2}} \to 0$.
   • Therefore, along this path, the limit is also $0$.

Conclusion:

Since the limit is $0$ along all the paths checked, it seems that the limit exists and equals $0$. However, the given answer says the limit does not exist. To confirm this, it would be necessary to check more paths or use polar coordinates for a more rigorous approach.

Polar Coordinates Approach:

In polar coordinates, $x = r \cos \theta$ and $y = r \sin \theta$, and $x^2 + y^2 = r^2$. The expression becomes:

$f(r, \theta) = \frac{(r \cos \theta)(r \sin \theta)}{\sqrt{r^2}} = \frac{r^2 \cos \theta \sin \theta}{r} = r \cos \theta \sin \theta$

As $r \to 0$, the expression $r \cos \theta \sin \theta \to 0$ regardless of $\theta$. Therefore, using polar coordinates also gives a limit of $0$.

Final Conclusion:

The limit is $0$, and it does exist. There might be a mistake in the provided answer.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is a general method for solving two-variable limits?
2. How do polar coordinates help in analyzing two-variable limits?
3. What are some cases where limits do not exist due to different path behaviors?
4. How would the result change if the denominator had a higher power of $x^2 + y^2$?
5. What is the significance of using specific paths like $y = mx$ to evaluate limits?

Tip: When checking limits of functions in multiple variables, always try different paths and consider using polar coordinates for a thorough analysis.