Find the limit lim⁡(x,y)→(0,0)x2−y2x2+y2lim(*x*,*y*)→(0,0)​*x*2+*y*2*x*2−*y*2​, if it exists.

To evaluate the limit

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

we can analyze the limit by approaching \((0, 0)\) along different paths. If the limit is the same along all paths, then it exists. Otherwise, it does not.

### Solution Steps
#### Step 1: Approach along \( y = 0 \)
If \( y = 0 \), then the expression becomes:
\[
\frac{x^2 - 0^2}{x^2 + 0^2} = \frac{x^2}{x^2} = 1.
\]
So, along the path \( y = 0 \), the limit is 1.

#### Step 2: Approach along \( x = 0 \)
If \( x = 0 \), then the expression becomes:
\[
\frac{0^2 - y^2}{0^2 + y^2} = \frac{-y^2}{y^2} = -1.
\]
Thus, along the path \( x = 0 \), the limit is -1.

### Conclusion
Since the limit depends on the path taken (it is 1 along \( y = 0 \) and -1 along \( x = 0 \)), the limit does not exist.

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### Related Questions
1. What if we approach along the line \( y = x \)?
2. How can polar coordinates be used to analyze this type of limit?
3. Why does taking different paths indicate the non-existence of a limit?
4. How would this problem change if we were only evaluating as \( x \to 0 \) or \( y \to 0 \) independently?
5. What role do symmetry and asymmetry play in determining multivariable limits?

#### Tip:
For limits in multiple dimensions, checking along various paths often quickly reveals if a limit does not exist.

Find the limit lim⁡(x,y)→(0,0)x^2−y^2/x^2+y^2, if it exists.

Explore the step-by-step solution for finding if the limit lim(x,y)→(0,0) (x^2 − y^2) / (x^2 + y^2) exists. This multivariable calculus problem examines path dependence and evaluates the limit along different paths. Suitable for college-level calculus students.

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