We are given the limit:

$\lim_{h \to 0} \frac{(2+h)^4 - 16}{h}$

This is a derivative expression, and we need to recognize it in the context of the definition of a derivative. The general form of the derivative of a function $f(x)$ at a point $x = a$ is:

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

By comparing this with the given expression, we can identify that this represents the derivative of some function $f(x)$ at $x = a$. Let's proceed with the following steps:

Step 1: Rewrite the given limit expression

We can observe that the limit is of the form:

$\frac{(2+h)^4 - 16}{h}$

This suggests that $f(x) = x^4$ because when we substitute $x = 2$ into $f(x) = x^4$, we get $f(2) = 2^4 = 16$.

Step 2: Check the derivative definition

The expression we are given seems to match the derivative definition for $f(x) = x^4$ at $x = 2$:

$f'(2) = \lim_{h \to 0} \frac{(2+h)^4 - 16}{h}$

Step 3: Confirm the function $f(x)$

The function is $f(x) = x^4$ because $(2+h)^4 - 16$ suggests we are evaluating the derivative of $x^4$ at $x = 2$.

Step 4: Determine the derivative

Now, we find the derivative of $f(x) = x^4$:

$f'(x) = 4x^3$

Thus, the derivative at $x = 2$ is:

$f'(2) = 4(2^3) = 4 \times 8 = 32$

Conclusion:

• The function is $f(x) = x^4$.
• The point where the derivative is evaluated is $x = 2$.

If you have any questions or would like more details, feel free to ask!

Related Questions:

1. What is the value of the derivative of $f(x) = x^4$ at $x = 3$?
2. How do we find the derivative of a polynomial function like $f(x) = x^n$?
3. What is the geometric interpretation of a derivative?
4. How do we compute limits involving powers and polynomials?
5. What happens to the limit if we change the expression to $\frac{(2+h)^3 - 8}{h}$?

Tip: To quickly identify the function in derivative problems, check the terms inside the limit expression and match it to the function's form (e.g., a binomial expansion or a simple power).