To solve the limit \( \lim_{x \to -2} (2x^4 - 3x^3 + 7x) \), you can substitute \( x = -2 \) directly into the expression since this is a polynomial function, which is continuous everywhere.

Let's evaluate the expression step by step:

\[
2x^4 - 3x^3 + 7x
\]

Substituting \( x = -2 \):

\[
2(-2)^4 - 3(-2)^3 + 7(-2)
\]

Now calculate each term:

\[
(-2)^4 = 16 \quad \text{so} \quad 2(-2)^4 = 2 \times 16 = 32
\]

\[
(-2)^3 = -8 \quad \text{so} \quad -3(-2)^3 = -3 \times (-8) = 24
\]

\[
7(-2) = -14
\]

Adding these values together:

\[
32 + 24 - 14 = 42
\]

Thus, the limit is:

\[
\lim_{x \to -2} (2x^4 - 3x^3 + 7x) = 42
\]

Would you like any more details or explanations?

Here are 5 related questions you might find interesting:
1. What are the properties of polynomial functions that allow direct substitution in limits?
2. How would the limit change if the function included a term like \( \frac{1}{x + 2} \)?
3. Can you evaluate the derivative of the function \( f(x) = 2x^4 - 3x^3 + 7x \) at \( x = -2 \)?
4. How does L'Hôpital's rule apply to limits involving indeterminate forms?
5. What happens to the limit if you approach a point where the function is not continuous?

**Tip:** When dealing with polynomial functions, direct substitution is always possible because polynomials are continuous everywhere.

Learn how to evaluate the limit \( \lim_{x \to -2} (2x^4 - 3x^3 + 7x) \) using direct substitution. This problem involves polynomial functions and continuity, suitable for advanced high school students.

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