Let's evaluate the limit of the function \( f(x, y) = x^2 + xy + y^2 \) as \((x, y)\) approaches \((0, 0)\) along the line \( y = mx \), where \( m \) is a constant.

### Substituting \( y = mx \) into the function

Substitute \( y = mx \) into the function \( f(x, y) \):

\[
f(x, mx) = x^2 + x(mx) + (mx)^2
\]

Simplifying this, we get:

\[
f(x, mx) = x^2 + mx^2 + m^2x^2 = (1 + m + m^2)x^2
\]

### Taking the limit as \( x \to 0 \)

Now, we find the limit of \( f(x, mx) \) as \( x \) approaches \( 0 \):

\[
\lim_{x \to 0} f(x, mx) = \lim_{x \to 0} (1 + m + m^2)x^2
\]

Since \( x^2 \) tends to \( 0 \) as \( x \) approaches \( 0 \), we have:

\[
\lim_{x \to 0} (1 + m + m^2)x^2 = 0
\]

### Conclusion

The limit of the function as \((x, y)\) approaches \((0, 0)\) along any line \( y = mx \) is \( 0 \). This confirms that the limit of \( f(x, y) \) as \((x, y)\) approaches \((0, 0)\) is indeed \( 0 \), consistent with the result for other paths we examined earlier.

Thus, the function \( f(x, y) \) is continuous at \((0, 0)\).

---

Would you like to dive deeper into this concept, or have any other questions? Here are five more questions related to this:

1. What does the limit tell us about the behavior of \( f(x, y) \) along different paths?
2. How can checking the limit along \( y = mx \) help confirm continuity at a point?
3. What happens if the limit depends on \( m \)? Does it imply discontinuity?
4. How can we apply similar techniques to check differentiability at a point?
5. How would the limit change if \( f(x, y) \) included a higher-order term, like \( x^4 \) or \( y^4 \)?

**Tip:** When evaluating limits for functions of two variables, exploring different paths (like \( y = mx \)) can reveal important insights into the function's behavior near a point.

Explore how to evaluate the limit of the function f(x, y) = x^2 + xy + y^2 as (x, y) approaches (0, 0) along the line y = mx. This problem delves into multivariable calculus, discussing limit definitions and theorems related to limits along paths. Suitable for undergraduate students studying calculus.

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