The limit below represents a derivative f’

The limit below represents a derivative f'(a). Find f(x) and a. lim h->0 = 6^3+h – 216 / h f(x)= a=

The Correct Answer and Explanation is:

Let’s carefully analyze the given limit:lim⁡h→063+h−216h\lim_{h \to 0} \frac{6^{3 + h} – 216}{h}h→0lim​h63+h−216​

This expression resembles the definition of a derivative, which is generally written as:f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a + h) – f(a)}{h}f′(a)=h→0lim​hf(a+h)−f(a)​

Comparing this with the provided limit:63+h−216h\frac{6^{3 + h} – 216}{h}h63+h−216​

It means:

• f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h
• f(a)=216f(a) = 216f(a)=216

First, find f(x)f(x)f(x).

Looking at f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h, this suggests f(x)=6xf(x) = 6^{x}f(x)=6x.

To confirm, let’s compute:

If f(x)=6xf(x) = 6^{x}f(x)=6x, then:f(a+h)=6a+hf(a + h) = 6^{a + h}f(a+h)=6a+hf(a)=6a=216f(a) = 6^{a} = 216f(a)=6a=216

We are told f(a)=216f(a) = 216f(a)=216, so:6a=2166^{a} = 2166a=216

We know that:63=2166^{3} = 21663=216

Thus:a=3a = 3a=3

Therefore:

• The function is f(x)=6xf(x) = 6^{x}f(x)=6x
• The value of a=3a = 3a=3

Final Answer:f(x)=6xanda=3f(x) = 6^{x} \quad \text{and} \quad a = 3f(x)=6xanda=3

Explanation

The given limit represents the definition of a derivative at a specific point. The general formula for the derivative of a function f(x)f(x)f(x) at x=ax = ax=a is:f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a + h) – f(a)}{h}f′(a)=h→0lim​hf(a+h)−f(a)​

In the provided expression:lim⁡h→063+h−216h\lim_{h \to 0} \frac{6^{3 + h} – 216}{h}h→0lim​h63+h−216​

We can match this with the derivative formula by recognizing f(a+h)=63+hf(a + h) = 6^{3 + h}f(a+h)=63+h and f(a)=216f(a) = 216f(a)=216. It implies that the original function f(x)f(x)f(x) must be f(x)=6xf(x) = 6^{x}f(x)=6x.

To verify, substitute x=ax = ax=a into the function:f(a)=6a=216f(a) = 6^{a} = 216f(a)=6a=216

We are asked to find the value of aaa such that 6a=2166^{a} = 2166a=216. Knowing that:63=6×6×6=36×6=2166^{3} = 6 \times 6 \times 6 = 36 \times 6 = 21663=6×6×6=36×6=216

So a=3a = 3a=3.

Therefore, the function f(x)=6xf(x) = 6^{x}f(x)=6x and the value a=3a = 3a=3 satisfy the given limit expression. This approach demonstrates how to connect a limit expression with the derivative definition to extract both the function and the evaluation point.

The correct function and value are:f(x)=6x,a=3f(x) = 6^{x}, \quad a = 3f(x)=6x,a=3