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To evaluate the limit \(\lim_{x \to -1} \left(2x^3 - 3x^2 + 2x + 5\right)\), we can directly substitute \(x = -1\) into the polynomial since polynomials are continuous everywhere.

Let's calculate:

\[
2(-1)^3 - 3(-1)^2 + 2(-1) + 5
\]

Now, let's evaluate each term:

\[
2(-1)^3 = 2(-1) = -2
\]
\[
-3(-1)^2 = -3(1) = -3
\]
\[
2(-1) = -2
\]
\[
5 \text{ remains as } 5
\]

So, substituting these into the expression:

\[
-2 - 3 - 2 + 5 = -7 + 5 = -2
\]

Thus, the limit is:

\[
\lim_{x \to -1} \left(2x^3 - 3x^2 + 2x + 5\right) = -2
\]

Would you like more details or have any other questions?

Here are some follow-up questions you might consider:

1. How can you determine if a function is continuous at a point?
2. What is the significance of continuity in evaluating limits?
3. What are other methods to evaluate limits besides direct substitution?
4. How would you evaluate a limit if the direct substitution led to an indeterminate form like \(\frac{0}{0}\)?
5. Can you explain why polynomials are continuous everywhere?

**Tip:** When evaluating limits of polynomials, direct substitution is often the easiest method because polynomials are continuous at all points.

Evaluate the limit: lim (2x^3 - 3x^2 + 2x + 5) as x approaches -1.

Learn how to evaluate the limit of the polynomial function 2x^3 - 3x^2 + 2x + 5 as x approaches -1 using direct substitution. This problem involves key calculus concepts and is suitable for Grades 11-12 or introductory calculus students.

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