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The given expression is the limit:

\[
\lim_{h \to 0} \frac{(7+h)^4 + h - 2401}{h}
\]

This resembles the definition of the derivative. To evaluate this, we can follow these steps:

### Step 1: Identify the function
From the structure, it looks like the limit is based on the derivative of some function evaluated at \( x = 7 \). The function that closely fits is:

\[
f(x) = x^4
\]

At \( x = 7 \), we have:

\[
f(7) = 7^4 = 2401
\]

So the limit expression involves computing the derivative of \( f(x) = x^4 \) at \( x = 7 \), with a small perturbation \( h \).

### Step 2: Apply the definition of the derivative
The derivative of \( f(x) = x^4 \) at any point \( x = a \) is defined as:

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

For \( f(x) = x^4 \), we are looking for \( f'(7) \), which means:

\[
f'(7) = \lim_{h \to 0} \frac{(7+h)^4 - 2401}{h}
\]

This simplifies to the form of the limit given, except the extra \( h \) term in the numerator. We can handle this next.

### Step 3: Expand and simplify
Let's expand \( (7 + h)^4 \) using the binomial theorem:

\[
(7+h)^4 = 7^4 + 4 \cdot 7^3 \cdot h + 6 \cdot 7^2 \cdot h^2 + 4 \cdot 7 \cdot h^3 + h^4
\]

This simplifies to:

\[
(7+h)^4 = 2401 + 1372h + 294h^2 + 28h^3 + h^4
\]

Now, substitute this into the original limit expression:

\[
\lim_{h \to 0} \frac{2401 + 1372h + 294h^2 + 28h^3 + h^4 + h - 2401}{h}
\]

Simplifying the numerator:

\[
= \lim_{h \to 0} \frac{1372h + 294h^2 + 28h^3 + h^4 + h}{h}
\]

### Step 4: Factor and simplify
Factor out \( h \) from the terms in the numerator:

\[
= \lim_{h \to 0} \frac{h(1372 + 294h + 28h^2 + h^3 + 1)}{h}
\]

Cancel \( h \) from the numerator and denominator:

\[
= \lim_{h \to 0} (1372 + 294h + 28h^2 + h^3 + 1)
\]

### Step 5: Take the limit
As \( h \to 0 \), the terms involving \( h \), \( h^2 \), and \( h^3 \) vanish, leaving:

\[
1372 + 1 = 1373
\]

Thus, the value of the limit is:

\[
\boxed{1373}
\]

---

Would you like further clarification or details on any step?

Here are 5 related questions to explore further:

1. What is the derivative of \( f(x) = x^4 \) at any general point \( x = a \)?
2. How does the binomial theorem work when expanding powers of binomials like \( (a + b)^n \)?
3. Can you explain why terms with higher powers of \( h \) vanish when taking limits?
4. How would the process change if the original function were \( f(x) = x^n \) for some other power \( n \)?
5. How do we interpret the derivative in the context of real-world problems?

**Tip**: In problems involving limits and derivatives, expanding using binomial theorem or Taylor series can simplify the process when small perturbations like \( h \) are involved.

Use the definition of the derivative to evaluate the limit: lim(h→0) [(7+h)^4 + h - 2401] / h.

Learn how to evaluate the limit lim(h→0) [(7+h)^4 + h - 2401] / h using the definition of the derivative. This problem involves expanding binomials and calculating the derivative of the function f(x) = x^4 at x = 7. Perfect for students studying calculus.

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