https://file.aiquickdraw.com/mathbot/file/17256686600472s6xcich.jpg

The given mathematical expression in the image is:

\[
\lim_{{x \to 3}} \frac{x^3 + 7x^2 - 29x}{x^2 - 4x - 21}
\]

To evaluate this limit, we should first simplify the expression. Let's start by factoring both the numerator and the denominator.

### Step 1: Factor the Numerator and Denominator

1. **Numerator: \(x^3 + 7x^2 - 29x\)**
   - Factor out an \(x\):  
   \[
   x(x^2 + 7x - 29)
   \]

   Now, we will check if \(x^2 + 7x - 29\) can be further factored, but it appears it cannot be factored easily.

2. **Denominator: \(x^2 - 4x - 21\)**
   - This can be factored as:  
   \[
   (x - 7)(x + 3)
   \]

### Step 2: Substitute the Factored Forms

The expression becomes:

\[
\lim_{{x \to 3}} \frac{x(x^2 + 7x - 29)}{(x - 7)(x + 3)}
\]

### Step 3: Plug in the Value \(x = 3\)

Now, substitute \(x = 3\) directly:

The denominator is \( (3 - 7)(3 + 3) = (-4)(6) = -24 \).

The numerator is \( 3(3^2 + 7 \cdot 3 - 29) = 3(9 + 21 - 29) = 3 \cdot 1 = 3 \).

Therefore, the limit simplifies to:

\[
\lim_{{x \to 3}} \frac{3}{-24} = -\frac{1}{8}
\]

### Final Answer

\[
\lim_{{x \to 3}} \frac{x^3 + 7x^2 - 29x}{x^2 - 4x - 21} = -\frac{1}{8}
\]

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

### Follow-Up Questions:

1. How can you factor cubic polynomials efficiently?
2. What are other techniques to solve limits that involve polynomials?
3. What if the denominator also has an indeterminate form at the limit point?
4. What does it mean when a limit does not exist?
5. How do you handle limits at infinity for similar rational functions?

#### Tip: Always check if direct substitution leads to an indeterminate form before simplifying further.

Evaluate the limit as x approaches 3 of the expression (x^3 + 7x^2 - 29x) / (x^2 - 4x - 21).

This detailed solution explains how to evaluate the limit as x approaches 3 of the polynomial expression (x^3 + 7x^2 - 29x) / (x^2 - 4x - 21). It covers key concepts such as polynomial factorization and limit evaluation techniques. Suitable for students in Grades 11-12.

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