To solve the limit 

\[
\lim_{x \to 0} \frac{(7 + 5x)^3 - 343}{x}
\]

we can use the limit definition and algebraic manipulation. This expression is in the form of a difference quotient that suggests using the binomial theorem or recognizing it as a derivative at a specific point. 

### Step-by-Step Solution

1. **Recognize the expression:**  
   Notice that \(343 = 7^3\), so we can rewrite the limit as:

   \[
   \lim_{x \to 0} \frac{(7 + 5x)^3 - 7^3}{x}
   \]

2. **Apply the binomial expansion (or recognize as a derivative):**  
   Expand \((7 + 5x)^3\) using the binomial theorem:

   \[
   (7 + 5x)^3 = 7^3 + 3 \cdot 7^2 \cdot (5x) + 3 \cdot 7 \cdot (5x)^2 + (5x)^3
   \]

   Simplify this:

   \[
   = 343 + 3 \cdot 49 \cdot 5x + 3 \cdot 7 \cdot 25x^2 + 125x^3
   \]

   \[
   = 343 + 735x + 525x^2 + 125x^3
   \]

3. **Subtract and simplify:**

   Now, substitute the expansion into the limit:

   \[
   \lim_{x \to 0} \frac{(343 + 735x + 525x^2 + 125x^3) - 343}{x}
   \]

   This simplifies to:

   \[
   \lim_{x \to 0} \frac{735x + 525x^2 + 125x^3}{x}
   \]

   Factor out an \(x\):

   \[
   \lim_{x \to 0} \left(735 + 525x + 125x^2\right)
   \]

4. **Evaluate the limit:**  

   As \(x \to 0\), the terms involving \(x\) vanish:

   \[
   = 735 + 0 + 0 = 735
   \]

### Conclusion

The value of the limit is:

\[
\lim_{x \to 0} \frac{(7 + 5x)^3 - 343}{x} = 735
\]

Would you like further clarification on any step or more details on binomial expansion?

### Related Questions

1. How does the binomial theorem work for expanding expressions?
2. How do you recognize the form of a derivative in a limit?
3. What is the geometric interpretation of this type of limit?
4. How can L'Hopital's Rule be applied to similar limits?
5. What are higher-order terms in a Taylor expansion?

**Tip:** Recognizing common forms, such as derivatives or binomial expansions, helps in simplifying limits effectively.

Learn how to solve the limit (7 + 5x)^3 - 343 / x as x approaches 0 with a detailed step-by-step solution. This problem involves applying the binomial theorem and recognizing the form of a derivative in a limit. Suitable for advanced high school and college-level students.

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