find the limit (x,y) to (0,0) of (tan(square root of(x^2 + y^2)))/(square root of(x^2 + y^2))

We are tasked with finding the limit of the following expression as \((x, y)\) approaches \((0, 0)\):

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

### Step 1: Polar coordinates
Since the expression involves \(x^2 + y^2\), it's natural to switch to polar coordinates:
- \(x = r \cos(\theta)\),
- \(y = r \sin(\theta)\),
where \(r = \sqrt{x^2 + y^2}\). 

In polar coordinates, the given expression becomes:

\[
\frac{\tan(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} = \frac{\tan(r)}{r}.
\]

Thus, we need to find the limit as \(r \to 0\) of:

\[
\lim_{r \to 0} \frac{\tan(r)}{r}.
\]

### Step 2: Apply the known limit
We know that:

\[
\lim_{r \to 0} \frac{\tan(r)}{r} = 1.
\]

This is a standard limit from calculus. Therefore, the limit of the original expression is:

\[
\lim_{(x, y) \to (0, 0)} \frac{\tan(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} = 1.
\]

### Final Answer:
\[
\boxed{1}.
\]

Would you like further details or explanations on any part of this solution?

Here are 5 related questions to expand your understanding:
1. What are polar coordinates, and how are they useful in solving limits?
2. Why is \(\lim_{r \to 0} \frac{\tan(r)}{r} = 1\) considered a standard limit?
3. How do we solve limits involving two variables like \((x, y)\) using polar coordinates?
4. Can you apply a similar method for \(\lim_{(x, y) \to (0, 0)} \frac{\sin(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}}\)?
5. What are the conditions for applying L'Hopital's Rule in finding limits?

**Tip:** Polar coordinates are extremely useful when dealing with limits involving expressions like \(\sqrt{x^2 + y^2}\), as they simplify the analysis by reducing two variables to one (the radius).

Find the limit of the expression (tan(sqrt(x^2 + y^2))) / (sqrt(x^2 + y^2)) as (x, y) approaches (0, 0). This solution uses polar coordinates and multivariable calculus concepts to arrive at the result of 1. Suitable for advanced high school or college-level students.

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