To evaluate the limit

$\lim_{(x, y) \to (0, 0)} y^2 \ln(|x|),$

we’ll analyze the behavior of $y^2 \ln(|x|)$ as $x \to 0$ and $y \to 0$ by considering different paths to see if the limit depends on the direction of approach or diverges.

1. Path $y = 0$:

   • Substitute $y = 0$ into the expression, giving $y^2 \ln(|x|) = 0 \cdot \ln(|x|) = 0$.
   • Along this path, the limit is $0$.

2. Path $y = x$:

   • Substitute $y = x$ into the expression to obtain: $y^2 \ln(|x|) = x^2 \ln(|x|).$
   • Now, as $x \to 0$, we need to consider the behavior of $x^2 \ln(|x|)$. Since $\ln(|x|) \to -\infty$ as $x \to 0$, $x^2 \ln(|x|)$ also tends to $0$, because $x^2$ approaches zero much faster than $\ln(|x|)$ diverges.

3. Path $y = x^{1/2}$:

   • Substitute $y = x^{1/2}$ into the expression, giving: $y^2 \ln(|x|) = (x^{1/2})^2 \ln(|x|) = x \ln(|x|).$
   • As $x \to 0$, $x \ln(|x|) \to 0$ for the same reason as above.

In each case, the expression $y^2 \ln(|x|)$ approaches $0$ as $(x, y) \to (0, 0)$, suggesting that the limit exists and is equal to $0$.

Thus, we conclude:

$\lim_{(x, y) \to (0, 0)} y^2 \ln(|x|) = 0.$

Would you like a deeper exploration of a specific path or have further questions?

Further Related Questions

1. How do you handle limits where variables approach zero but involve logarithmic functions?
2. What other methods could confirm this limit result more rigorously?
3. Why does $x^2$ tend to zero faster than $\ln(|x|)$ diverges?
4. How can polar coordinates help in evaluating multivariable limits?
5. Can this approach be generalized to functions of three or more variables?

Tip:

When calculating multivariable limits, always test different paths to ensure consistency across approaches.