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Let's solve the given limit step-by-step.

### Problem
We are solving the following:
\[
\lim_{x \to 0} \left( \frac{\sin(7x)}{9x} + (x^2 + 1) \cdot \frac{\sin(9x)}{\sin(7x)} \right)
\]

---

### Step 1: Simplify the first term
Using the small-angle approximation \(\sin(kx) \approx kx\) as \(x \to 0\), the first term becomes:
\[
\frac{\sin(7x)}{9x} \to \frac{7x}{9x} = \frac{7}{9}.
\]

---

### Step 2: Simplify the second term
The second term is \((x^2 + 1) \cdot \frac{\sin(9x)}{\sin(7x)}\).

#### (a) Simplify \(\frac{\sin(9x)}{\sin(7x)}\):
Using \(\sin(kx) \approx kx\),
\[
\frac{\sin(9x)}{\sin(7x)} \to \frac{9x}{7x} = \frac{9}{7}.
\]

#### (b) Evaluate \(x^2 + 1\) as \(x \to 0\):
\[
x^2 + 1 \to 1.
\]

Thus, the second term becomes:
\[
(x^2 + 1) \cdot \frac{\sin(9x)}{\sin(7x)} \to 1 \cdot \frac{9}{7} = \frac{9}{7}.
\]

---

### Step 3: Combine the results
Adding the two terms:
\[
\lim_{x \to 0} \left( \frac{\sin(7x)}{9x} + (x^2 + 1) \cdot \frac{\sin(9x)}{\sin(7x)} \right) = \frac{7}{9} + \frac{9}{7}.
\]

---

### Step 4: Simplify the sum
To combine \(\frac{7}{9}\) and \(\frac{9}{7}\), find a common denominator:
\[
\frac{7}{9} + \frac{9}{7} = \frac{49}{63} + \frac{81}{63} = \frac{130}{63}.
\]

Thus, the answer is:
\[
\boxed{\frac{130}{63}}
\]

---

Would you like further clarification or a deeper breakdown?  
Here are related questions to explore:

1. How does the small-angle approximation work for trigonometric functions?
2. Can you generalize the approach to similar trigonometric limits?
3. Why is it valid to assume \(x^2 + 1 \approx 1\) as \(x \to 0\)?
4. How can L'Hôpital's Rule be used to solve such limits?
5. What are common mistakes when simplifying fractions in such problems?

**Tip:** Always verify trigonometric approximations when \(x \to 0\) using Taylor series for better precision.

Calculate the value of the limit: lim x→0 (sin(7x)/(9x) + (x^2 + 1) * (sin(9x)/sin(7x))). Provide your answer as a fraction in lowest terms.

Learn how to calculate the limit lim x→0 (sin(7x)/(9x) + (x^2 + 1) * (sin(9x)/sin(7x))) with a detailed step-by-step solution. This problem involves small-angle trigonometric approximations and fraction simplification, suitable for advanced high school students in Grades 11-12.

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