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The given expression is:

\[
\lim_{{x \to 3}} (4x^2 - 2x - 6)
\]

To find this limit, we can directly substitute \(x = 3\) into the expression because it is a polynomial, and polynomials are continuous functions. Let's calculate step by step:

1. Substitute \(x = 3\) into the expression:
\[
4(3)^2 - 2(3) - 6
\]

2. Simplify each term:
\[
4(9) - 2(3) - 6 = 36 - 6 - 6
\]

3. Final result:
\[
36 - 6 - 6 = 24
\]

Thus, the limit is:

\[
\lim_{{x \to 3}} (4x^2 - 2x - 6) = 24
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:
1. How does the limit change if we have a different point of convergence, say \( x \to 2 \)?
2. What happens if the polynomial had a denominator that causes division by zero?
3. How would you handle the limit if there was an absolute value involved in the expression?
4. What if the function was piecewise defined? How would the limit be approached at a break point?
5. Can you explain the concept of continuity and how it relates to limits?

**Tip:** When working with polynomial limits, direct substitution is always a good first approach because polynomials are continuous everywhere.

Find the limit of the polynomial expression lim(x→3)(4x²−2x−6). Learn how to apply direct substitution to calculate limits in polynomial functions. This solution demonstrates key algebraic concepts and the continuity theorem.

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