The given expression:

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

resembles the definition of the derivative:

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

In this case, we can identify $f(x)$ and $a$ by comparing the structure of the limit expression.

1. Finding $f(x)$: Notice that the term $(4 + h)^2$ suggests that $f(x) = x^2$ because the expression $(4 + h)^2$ matches the form $f(4 + h)$ for $f(x) = x^2$.

2. Finding $a$: Since the expression is evaluated at $4 + h$, it indicates that $a = 4$.

So:

• $f(x) = x^2$
• $a = 4$

This setup allows us to express the limit as the derivative of $f(x) = x^2$ at $x = 4$.

Let me know if you'd like more details, or if you have any questions!

Related Questions:

1. How would you evaluate the limit if $f(x) = x^3$ and $a = 4$?
2. What would change if the function $f(x)$ was $\sqrt{x}$ and $a = 4$?
3. How does this limit expression change if we evaluate $f(x) = x^2$ at $a = 5$?
4. What is the purpose of the derivative definition in calculus?
5. How can you confirm $f(x)$ and $a$ from similar derivative limit expressions?

Tip: To solve these types of problems, always compare the expression with the general derivative formula $f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h}$ to identify the function $f(x)$ and point $a$.