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The problem asks you to evaluate the limit using the definition of the derivative:

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

The general form of the definition of the derivative is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Here, we can observe that \( f(x) \) is a cubic function evaluated at \( x = 7 \), and we are trying to evaluate the derivative at that point. Let's rewrite and evaluate step by step:

### Step 1: Define the Function
Let's define \( f(x) = x^3 \).

Thus, \( f(7) = 7^3 = 343 \).

### Step 2: Substitute into the Limit
The given limit expression resembles the definition of a derivative, with \( f(x) = x^3 \). Let’s rewrite the limit in terms of \( f \):

\[
\lim_{h \to 0} \frac{f(7+h) - f(7)}{h}
\]

### Step 3: Substitute the Function
\[
f(7+h) = (7+h)^3
\]

So, we have:

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

### Step 4: Expand \( (7+h)^3 \)
We need to expand \( (7+h)^3 \):

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

\[
(7+h)^3 = 343 + 147h + 21h^2 + h^3
\]

### Step 5: Substitute Back into the Limit
Now substitute back into the limit:

\[
\lim_{h \to 0} \frac{343 + 147h + 21h^2 + h^3 - 343}{h}
\]

### Step 6: Simplify
\[
\lim_{h \to 0} \frac{147h + 21h^2 + h^3}{h}
\]

Factor \( h \) out from the numerator:

\[
\lim_{h \to 0} \frac{h(147 + 21h + h^2)}{h}
\]

Cancel \( h \):

\[
\lim_{h \to 0} (147 + 21h + h^2)
\]

### Step 7: Evaluate the Limit
As \( h \to 0 \):

\[
147 + 21h + h^2 \to 147
\]

### Final Answer
\[
\lim_{h \to 0} \frac{(7+h)^3 - 343}{h} = 147
\]

Thus, the value of the limit is:

\[
\boxed{147}
\]

Would you like more details on this problem, or do you have any further questions?

Here are 5 related questions you could explore:
1. How does the definition of the derivative apply to other functions, like trigonometric functions?
2. What are the steps to expand binomials of higher powers, like \((x + h)^4\)?
3. How can we use this method to find the derivative of other polynomial functions?
4. What happens if we apply a similar limit to a function that is not differentiable at the point?
5. How can we use the chain rule to differentiate more complex compositions of functions?

**Tip:** When applying the limit definition of the derivative, expanding polynomials and carefully simplifying is key to identifying the derivative value accurately.

Use the definition of the derivative to evaluate the limit: lim(h → 0) ((7 + h)^3 + h - 343) / h

Learn how to evaluate the limit lim(h → 0) [(7+h)^3 - 343] / h using the definition of the derivative. This step-by-step guide covers the binomial expansion and the application of derivative rules, suitable for students in advanced calculus classes.

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