Which function $f\left(x\right)$ and value $a$ correspond to the following limit representing the derivative of $f$ at $a$?$$\underset{h\to 0}{lim}\frac{{(3+h)}^2-9}{h}$$

• A. $f\left(x\right)=x^2$, $a=3$
• B. $f\left(x\right)=x^3$, $a=2$
• C. $f\left(x\right)=2x$, $a=3$
• D. $f\left(x\right)=x^2+1$, $a=3$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the definition of the derivative. The derivative of a function f(x) at a point a is given by the limit: lim_{h \(\to\) 0} \(\frac{f(a + h) - f(a)}{h}\). This matches the structure of the given limit.
2. Step 2: Compare the given limit expression lim_{h \(\to\) 0} \(\frac{(3 + h)^2 - 9}{h}\) with the derivative definition. Here, f(a + h) corresponds to (3 + h)^2, and f(a) corresponds to 9.
3. Step 3: Identify the function f(x). Since f(a) = 9 and a = 3, we can deduce that f(x) = x^2 because (3)^2 = 9.
4. Step 4: Verify the value of a. The expression (3 + h)^2 represents f(a + h), which confirms that a = 3.
5. Step 5: Conclude that the function and value corresponding to the limit are f(x) = x^2 and a = 3.