https://file.aiquickdraw.com/mathbot/file/1727368301677lgxmpvrc.jpg

To solve the limit

\[
\lim_{x \to 1} \frac{x - x^4}{1 - x},
\]

we need to simplify the expression to determine what should go into the first box. Let's break it down:

1. **Factor the numerator**: The numerator is \( x - x^4 \), and we can factor it as:

   \[
   x - x^4 = x(1 - x^3) = x(1 - (x)(x^2)) = x(1 - x)(1 + x + x^2).
   \]

2. **Simplify the expression**: Now we can substitute this factored form back into the original expression:

   \[
   \frac{x(1 - x)(1 + x + x^2)}{1 - x}.
   \]

   The \(1 - x\) terms cancel out, leaving us with:

   \[
   \lim_{x \to 1} x(1 + x + x^2).
   \]

3. **Evaluate the limit**: We now evaluate the limit as \( x \to 1 \):

   \[
   1(1 + 1 + 1^2) = 1(1 + 1 + 1) = 1 \times 3 = 3.
   \]

Therefore, the first box should contain \( x(1 + x + x^2) \), and the final answer for the limit is \( 3 \).

Let me know if you'd like further clarification or additional details!

---

### Relative Questions:
1. What are common techniques to simplify limits involving polynomials?
2. How do you factor higher-degree polynomials, such as \( x^4 \)?
3. What role does the cancellation of terms play in evaluating limits?
4. How can you verify that a limit results in a removable discontinuity?
5. What is the importance of limits in calculus?

#### Tip: When you have an indeterminate form in limits (like \( \frac{0}{0} \)), try factoring or applying L'Hopital's Rule.

This problem demonstrates how to evaluate a limit involving a polynomial. Learn how to factor the expression x - x^4 and simplify the limit lim_{x -> 1} (x - x^4) / (1 - x). Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation