https://r2.erweima.ai/mathbot/image/172619786639200itb6hq.jpg

To solve the given limit:

\[
\lim_{{x \to 1}} \frac{{x^2 + x - 2}}{{x^2 + 2x - 3}}
\]

### Step-by-step solution:

1. **Factor the numerator and the denominator**:
   
   - The numerator \(x^2 + x - 2\) can be factored as:
     \[
     x^2 + x - 2 = (x - 1)(x + 2)
     \]

   - The denominator \(x^2 + 2x - 3\) can be factored as:
     \[
     x^2 + 2x - 3 = (x - 1)(x + 3)
     \]

2. **Rewrite the expression**:
   After factoring, the expression becomes:
   \[
   \frac{{(x - 1)(x + 2)}}{{(x - 1)(x + 3)}}
   \]

3. **Simplify the expression**:
   Cancel the common factor \((x - 1)\), assuming \(x \neq 1\) (since we are considering the limit as \(x \to 1\), not at \(x = 1\)):
   \[
   \frac{{x + 2}}{{x + 3}}
   \]

4. **Evaluate the limit**:
   Now, substitute \(x = 1\) into the simplified expression:
   \[
   \frac{{1 + 2}}{{1 + 3}} = \frac{3}{4}
   \]

Thus, the value of the limit is:

\[
\boxed{\frac{3}{4}}
\]

Would you like further clarification on any step?

### Related questions:
1. What happens if the denominator cannot be factored easily?
2. How would you approach a limit where direct substitution leads to infinity?
3. Can limits always be simplified by factoring?
4. What role does L'Hopital's Rule play in solving limits?
5. What if the limit were to approach a different number like \(x \to 0\)?

**Tip:** When factoring polynomials, always check for common factors or apply quadratic formulas to simplify expressions before evaluating limits.

Calculate the limit: lim (x -> 1) ((x^2 + x - 2) / (x^2 + 2x - 3))

Learn how to calculate the limit of the expression (x^2 + x - 2) / (x^2 + 2x - 3) as x approaches 1. This step-by-step guide includes factoring of quadratic equations and simplification techniques, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation