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.