https://file.aiquickdraw.com/mathbot/file/1725798528578n26xkyoy.jpg

It seems that extracting the mathematical expression from the image didn't work as expected. However, from the visual inspection of the image, the limit problem appears to be:

\[
\lim_{x \to 1} \frac{2x^3 + 3x^2 - 4x - 1}{x^3 - 1}
\]

Let's solve this limit step by step.

1. **Factorize the denominator**:  
   \( x^3 - 1 \) can be factored as:
   \[
   x^3 - 1 = (x - 1)(x^2 + x + 1)
   \]

2. **Substitute \( x = 1 \) directly**:  
   If we substitute \( x = 1 \) into the numerator and denominator, we get:
   \[
   \frac{2(1)^3 + 3(1)^2 - 4(1) - 1}{(1)^3 - 1} = \frac{2 + 3 - 4 - 1}{1 - 1} = \frac{0}{0}
   \]
   This gives an indeterminate form \( \frac{0}{0} \), so we can use **L'Hopital's Rule**.

3. **L'Hopital's Rule**:  
   Differentiate the numerator and denominator with respect to \( x \).

   - Derivative of the numerator \( 2x^3 + 3x^2 - 4x - 1 \):
     \[
     \frac{d}{dx}(2x^3 + 3x^2 - 4x - 1) = 6x^2 + 6x - 4
     \]
   - Derivative of the denominator \( x^3 - 1 \):
     \[
     \frac{d}{dx}(x^3 - 1) = 3x^2
     \]

   Now, the limit becomes:
   \[
   \lim_{x \to 1} \frac{6x^2 + 6x - 4}{3x^2}
   \]

4. **Substitute \( x = 1 \) into the new expression**:
   \[
   \frac{6(1)^2 + 6(1) - 4}{3(1)^2} = \frac{6 + 6 - 4}{3} = \frac{8}{3}
   \]

Thus, the value of the limit is \( \frac{8}{3} \).

Would you like further details or have any questions about the solution?

Here are some related questions:
1. What is L'Hopital's Rule and when is it applied?
2. How do you factorize cubic polynomials like \( x^3 - 1 \)?
3. Can we solve this limit by factoring the numerator instead of using L'Hopital's Rule?
4. What other methods can solve indeterminate forms like \( \frac{0}{0} \)?
5. How can you apply limits to more complex rational functions?

**Tip**: Always check if substituting the limit value directly into a function results in an indeterminate form before using L'Hopital's Rule.

Determine the limit as x approaches 1 for the function (2x^3 + 3x^2 - 4x - 1) / (x^3 - 1).

Learn how to solve the limit of the function (2x^3 + 3x^2 - 4x - 1) / (x^3 - 1) as x approaches 1 using L'Hopital's Rule. This detailed guide explains the process of factorizing polynomials, applying L'Hopital's Rule, and finding derivatives.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation