Math Problem Statement

Can you answer this?

Can you answer this?

Solution

Certainly! Let's solve the limit:
limx1x2+2x3x23x+2.\lim_{x \to 1} \frac{x^2 + 2x - 3}{x^2 - 3x + 2}.


Step 1: Factorize the numerator and denominator

Numerator:
x2+2x3x^2 + 2x - 3
Factorizing: x2+2x3=(x+3)(x1).x^2 + 2x - 3 = (x + 3)(x - 1).

Denominator:
x23x+2x^2 - 3x + 2
Factorizing: x23x+2=(x2)(x1).x^2 - 3x + 2 = (x - 2)(x - 1).


Step 2: Rewrite the expression

After factorization, the function becomes: x2+2x3x23x+2=(x+3)(x1)(x2)(x1).\frac{x^2 + 2x - 3}{x^2 - 3x + 2} = \frac{(x + 3)(x - 1)}{(x - 2)(x - 1)}.


Step 3: Cancel common terms

The factor x1x - 1 appears in both the numerator and denominator, so we cancel it. This simplifies the expression to: x+3x2,for x1.\frac{x + 3}{x - 2}, \quad \text{for } x \neq 1.


Step 4: Evaluate the limit as x1x \to 1

Substitute x=1x = 1 into the simplified expression: x+3x21+312=41=4.\frac{x + 3}{x - 2} \to \frac{1 + 3}{1 - 2} = \frac{4}{-1} = -4.


Final Answer:

limx1x2+2x3x23x+2=4.\lim_{x \to 1} \frac{x^2 + 2x - 3}{x^2 - 3x + 2} = -4.


Would you like a detailed explanation of any step?
Here are some related questions to deepen your understanding:

  1. What is the importance of factoring in solving limits?
  2. Why do we cancel common terms, and what happens if we don't?
  3. How would the limit change if the denominator had no common factor with the numerator?
  4. What does the behavior of the function around x=1x = 1 tell us about the graph?
  5. Can we use L'Hôpital's Rule to solve this limit, and how would it work?

Tip: Always check for factorization when solving limits involving polynomials to simplify the problem!

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Math Problem Analysis

Mathematical Concepts

Limits
Polynomial Factorization
Rational Functions

Formulas

Factorization of polynomials
Simplification of rational expressions

Theorems

Limit Laws
Factorization techniques

Suitable Grade Level

Grades 10-12