$$ \lim_{x\rarr0}\,\frac{(x+\Delta x)^2-3(x+\Delta x)+9-(x^2-3x+9)}{\Delta x} $$

Let's evaluate the given limit:

\[
\lim_{\Delta x \to 0}\, \frac{(x+\Delta x)^2 - 3(x+\Delta x) + 9 - (x^2 - 3x + 9)}{\Delta x}
\]

### Step 1: Expand the terms
First, we expand the terms inside the numerator:

\[
(x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2
\]
\[
-3(x + \Delta x) = -3x - 3\Delta x
\]

Now substitute these into the numerator:

\[
\left( x^2 + 2x\Delta x + (\Delta x)^2 - 3x - 3\Delta x + 9 \right) - \left( x^2 - 3x + 9 \right)
\]

### Step 2: Simplify the expression
Let's cancel out common terms. Notice that $x^2 - 3x + 9$ appears in both parts, so they cancel:

\[
x^2 + 2x\Delta x + (\Delta x)^2 - 3x - 3\Delta x + 9 - x^2 + 3x - 9
\]
\[
= 2x\Delta x + (\Delta x)^2 - 3\Delta x
\]

### Step 3: Factor out $\Delta x$
Now factor $\Delta x$ from the remaining terms:

\[
\Delta x(2x + \Delta x - 3)
\]

### Step 4: Apply the limit
Now, we apply the limit as $\Delta x \to 0$:

\[
\lim_{\Delta x \to 0}\, \frac{\Delta x(2x + \Delta x - 3)}{\Delta x} = \lim_{\Delta x \to 0}\, (2x + \Delta x - 3)
\]

Since $\Delta x$ approaches 0, this simplifies to:

\[
2x - 3
\]

### Final answer:
\[
\lim_{\Delta x \to 0}\, \frac{(x+\Delta x)^2-3(x+\Delta x)+9-(x^2-3x+9)}{\Delta x} = 2x - 3
\]

Would you like more details or have any questions?

Here are some related questions:
1. How does the derivative relate to this limit calculation?
2. What happens if we change the function inside the limit?
3. How would the result change if we used higher-order polynomials?
4. What are the physical interpretations of derivatives in real-world applications?
5. How do we apply the chain rule in derivative calculations?

**Tip:** When taking limits involving differences, always look to cancel terms to simplify the expression before applying the limit.

$$ \lim_{x \to 0} \frac{(x+\Delta x)^2-3(x+\Delta x)+9-(x^2-3x+9)}{\Delta x} $$

Learn how to calculate the limit $$ \lim_{\Delta x \to 0} \frac{(x+\Delta x)^2 - 3(x+\Delta x) + 9 - (x^2 - 3x + 9)}{\Delta x} $$ with step-by-step expansion, simplification, and applying the limit definition of derivatives. Ideal for calculus students in Grades 11-12.

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