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.