To solve the limit of \(\frac{x-1}{x+1}\) as \(x\) approaches \(-1\), we follow these steps:

1. **Direct Substitution**:
   Substitute \(x = -1\) into the expression:

   \[
   \frac{-1 - 1}{-1 + 1} = \frac{-2}{0}
   \]

   This results in a division by zero, indicating that we need further analysis.

2. **Behavior Analysis**:
   Analyze the behavior of the function as \(x\) approaches \(-1\) from both the left and the right sides.

   - As \(x\) approaches \(-1\) from the left (\(x \to -1^-\)):
     The numerator \((x-1)\) approaches \(-2\), and the denominator \((x+1)\) approaches \(0\) from the negative side. Thus, \(\frac{x-1}{x+1}\) approaches \(+\infty\).

   - As \(x\) approaches \(-1\) from the right (\(x \to -1^+\)):
     The numerator \((x-1)\) approaches \(-2\), and the denominator \((x+1)\) approaches \(0\) from the positive side. Thus, \(\frac{x-1}{x+1}\) approaches \(-\infty\).

3. **Conclusion**:
   Since the limits from the left and the right are not equal, the limit does not exist.

Thus, the limit of \(\frac{x-1}{x+1}\) as \(x\) approaches \(-1\) does not exist.

Would you like more details or have any questions?

Here are some more questions you might be interested in:

1. How to evaluate the limit of a function as \(x\) approaches infinity?
2. How to find the derivative of a given function?
3. How to integrate a given function?
4. What is the Taylor series expansion of a function?
5. How to solve a differential equation?

**Tip:** When evaluating limits that lead to indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), consider using L'Hôpital's Rule.

Learn how to solve the limit of (x-1)/(x+1) as x approaches -1 with detailed step-by-step analysis. This problem involves understanding limits, indeterminate forms, and application of limit laws. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation