lim┬((x,y)→(0,1) )⁡〖(x^3+ y^2)/(x^2+2)〗 

To evaluate the limit

\[
\lim_{(x,y) \to (0,1)} \frac{x^3 + y^2}{x^2 + 2},
\]

we need to consider the behavior of the function as \((x, y)\) approaches \((0, 1)\).

1. **Substitute \(y = 1\)**: As \((x, y) \to (0, 1)\), we can fix \(y = 1\) and examine the behavior as \(x \to 0\).

   \[
   \lim_{x \to 0} \frac{x^3 + 1^2}{x^2 + 2} = \lim_{x \to 0} \frac{x^3 + 1}{x^2 + 2}.
   \]

   When \(x = 0\):

   \[
   \frac{0^3 + 1}{0^2 + 2} = \frac{1}{2}.
   \]

2. **Consider \(x = 0\) directly**: If we substitute \(x = 0\) and let \(y \to 1\):

   \[
   \frac{0^3 + y^2}{0^2 + 2} = \frac{y^2}{2}.
   \]

   As \(y \to 1\):

   \[
   \frac{1}{2}.
   \]

In both cases, the expression approaches \(\frac{1}{2}\) as \((x, y)\) approaches \((0, 1)\).

Therefore, the limit is

\[
\lim_{(x,y) \to (0,1)} \frac{x^3 + y^2}{x^2 + 2} = \frac{1}{2}.
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you evaluate limits involving two variables along different paths?
2. What conditions ensure that a two-variable limit exists?
3. How does L'Hopital's Rule apply to multivariable limits?
4. What happens if the limit yields different values along different paths?
5. How can polar coordinates simplify evaluating multivariable limits?

**Tip:** Always consider different paths when evaluating multivariable limits to ensure consistency in the limit's value.

Learn how to evaluate the limit of the function (x^3 + y^2)/(x^2 + 2) as the point (x, y) approaches (0, 1). Detailed step-by-step solution provided, considering different paths and substitutions. Suitable for college-level students and those studying multivariable calculus.

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