lim
(௫,௬)→(ଵ;ଵ) ቆ
𝑥
ଶ − 1
𝑥 − 1 +
𝑦 + 1
𝑦
ଶ − 1ቇ =

Let's evaluate the limit as \((x, y) \to (1, 1)\) of the expression:

\[
\lim_{(x, y) \to (1, 1)} \left(\frac{x^2 - 1}{x - 1} + \frac{y + 1}{y^2 - 1}\right)
\]

### Step 1: Simplify the expression
First, let's simplify the two fractions:

1. **For** \(\frac{x^2 - 1}{x - 1}\):
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]
   So,
   \[
   \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1 \quad \text{(for } x \neq 1\text{)}
   \]

2. **For** \(\frac{y + 1}{y^2 - 1}\):
   \[
   y^2 - 1 = (y - 1)(y + 1)
   \]
   So,
   \[
   \frac{y + 1}{y^2 - 1} = \frac{y + 1}{(y - 1)(y + 1)} = \frac{1}{y - 1} \quad \text{(for } y \neq 1\text{)}
   \]

### Step 2: Substitute and evaluate the limit
The expression now simplifies to:

\[
\lim_{(x, y) \to (1, 1)} \left( (x + 1) + \frac{1}{y - 1} \right)
\]

### Step 3: Evaluate the limit as \((x, y) \to (1, 1)\)

- \(x + 1\) as \(x \to 1\) is \(1 + 1 = 2\).
- \(\frac{1}{y - 1}\) as \(y \to 1\) tends to infinity (since \(\frac{1}{y - 1}\) has a singularity at \(y = 1\)).

Thus, the limit does not exist in the conventional sense because the term \(\frac{1}{y - 1}\) tends to infinity as \(y\) approaches 1.

### Final Answer:
\[
\lim_{(x, y) \to (1, 1)} \left(\frac{x^2 - 1}{x - 1} + \frac{y + 1}{y^2 - 1}\right) = \infty
\]

Would you like more details on this process, or do you have any questions?

---

Here are some related questions:

1. What is the significance of indeterminate forms in limits?
2. How do you evaluate limits involving infinity?
3. What are the conditions under which a limit does not exist?
4. How do singularities affect the calculation of limits?
5. Can you explain the concept of continuity in multivariable functions?

**Tip:** When evaluating limits involving fractions, always consider simplifying expressions to reveal potential indeterminate forms or singularities.

Learn how to evaluate the limit as (x, y) approaches (1, 1) of the expression (x^2 - 1)/(x - 1) + (y + 1)/(y^2 - 1). This problem involves understanding limits involving indeterminate forms and singularities. Suitable for advanced undergraduate students and those interested in advanced calculus concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation