To find the limit of the function \(3x^2 + 9x - 5\) as \(x\) approaches 6, we can directly substitute the value of \(x = 6\) into the function since it is a polynomial and continuous everywhere.

\[
\lim_{x \to 6} (3x^2 + 9x - 5)
\]

Substitute \(x = 6\) into the expression:

\[
= 3(6)^2 + 9(6) - 5
\]

Calculate the values:

\[
= 3(36) + 54 - 5
\]

\[
= 108 + 54 - 5
\]

\[
= 162 - 5 = 157
\]

So, the limit is \( \boxed{157} \).

Would you like any additional details or have any questions about this? Here are five related questions you might find interesting:

1. How would you find the limit if the function had a discontinuity at \(x = 6\)?
2. What is the derivative of the function \(3x^2 + 9x - 5\)?
3. How does the limit change if the function is modified to \(3x^2 + 9x + 5\)?
4. What is the general process for finding limits of more complex functions?
5. How does L'Hôpital's rule apply when finding limits that result in indeterminate forms?

**Tip:** For polynomial functions, the limit as \(x\) approaches any real number is simply the value of the polynomial at that point, since polynomials are continuous everywhere.

Learn how to find the limit of the polynomial 3x^2 + 9x - 5 as x approaches 6. Step-by-step solution provided with calculations and explanations. Suitable for high school students studying limits and polynomial functions.

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