Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(x)=3 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=3\) is \(0\).

Step 1: Write down the function

Our function is simply \(f(x) = 3\), a horizontal line at \(y = 3\).

Step 2: Write down the limit definition of the derivative

The limit definition of the derivative, also known as the formal definition or the difference quotient, is: \[f'(x) = \lim_{h\rightarrow0} [f(x+h)-f(x)]/h\]

Step 3: Substitute the function into the limit definition

Using \(f(x) = 3\), the limit definition becomes: \[f'(x) = \lim_{h\rightarrow0} [(3)-(3)]/h = \lim_{h\rightarrow0} [0]/h\]

Step 4: Evaluate the limit

As \(h\) approaches zero, \(0/h\) is always \(0\). Therefore the limit is \(0\).