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

To solve the limit 

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

we need to evaluate the expression as \( (x,y) \) approaches the point \( (0,1) \).

### Step 1: Substitute \( y = 1 \) (since \( y \) approaches 1)
First, substitute \( y = 1 \) into the expression, as \( y \) approaches 1:
\[
\frac{x^3 + 1^2}{x^2 + 1} = \frac{x^3 + 1}{x^2 + 1}.
\]

### Step 2: Evaluate the limit as \( x \) approaches 0
Now, find the limit as \( x \) approaches 0:
\[
\lim_{x \to 0} \frac{x^3 + 1}{x^2 + 1}.
\]
Substitute \( x = 0 \) into the expression:
\[
\frac{0^3 + 1}{0^2 + 1} = \frac{1}{1} = 1.
\]

Thus, the limit is 

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

Would you like more details on this solution or have any questions?

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Here are five related questions you might be interested in:

1. How do you solve limits involving multiple variables?
2. What are the common methods to evaluate limits in calculus?
3. How do you evaluate limits that approach infinity?
4. What is the epsilon-delta definition of a limit in multivariable calculus?
5. How can you determine if a limit does not exist?

**Tip:** When evaluating limits, always check if substituting the limit point directly into the function yields an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). If it does, consider using methods like factoring, rationalization, or L'Hôpital's rule to resolve it.

Learn how to evaluate the limit of the expression (x^3 + y^2) / (x^2 + 1) as the point (x,y) approaches (0,1). Follow a step-by-step solution using substitution and limit evaluation methods. Suitable for college-level students studying multivariable calculus.

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