https://file.aiquickdraw.com/mathbot/file/17263046043422fndo4k8.jpg

The image shows a limit evaluation problem:

\[
\lim_{{x \to -1}} \left( 2x^3 - 4x^2 + 1 \right)
\]

The steps written are as follows:

1. Substitute \( x = -1 \) directly into the expression:
   \[
   2(-1)^3 - 4(-1)^2 + 1
   \]

2. Simplify the terms:
   \[
   2(-1) - 4(1) + 1
   \]

   Which results in:
   \[
   -2 - 4 + 1
   \]

3. Finally:
   \[
   -2 - 4 + 1 = -5
   \]

Thus, the evaluated limit is:
\[
\boxed{-5}
\]

Let me know if you need further details or clarification!

Here are 5 relative questions to expand on this:

1. How do you find the limit when direct substitution results in an indeterminate form like \(\frac{0}{0}\)?
2. What is the difference between one-sided and two-sided limits?
3. How can L'Hôpital's Rule be used to solve limits involving indeterminate forms?
4. What are the steps to apply the Squeeze Theorem for finding limits?
5. How can limits be used to find the derivative of a function?

**Tip**: Always check if direct substitution gives a real value before attempting more complex methods like factoring or L'Hôpital's Rule for evaluating limits.

Evaluate the limit lim(x -> -1) (2x^3 - 4x^2 + 1)

Learn how to evaluate the limit of the polynomial function 2x^3 - 4x^2 + 1 as x approaches -1 using direct substitution. This step-by-step guide covers key concepts such as polynomial limits and substitution techniques. Suitable for students in Grades 10-12.

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