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.