Question

Each limit represents the derivative of some function f at some number a. State such as an f and a in each case.

\(\mathop {{\bf{lim}}}\limits_{x \to \frac{{\bf{1}}}{{\bf{4}}}} \frac{{\frac{{\bf{1}}}{x} - {\bf{4}}}}{{x - \frac{{\bf{1}}}{{\bf{4}}}}}\)

Short answer

The function is \(f\left( x \right) = \frac{1}{x}\) and \(a = \frac{1}{4}\).

Step-by-step solution

Step 1:Write the given limit as in Definition 4

According to definition 5, the derivative of a function f is given by the equation:

\(f'\left( a \right) = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\)

The expression \(\mathop {\lim }\limits_{x \to \frac{1}{4}} \frac{{\frac{1}{x} - 4}}{{x - \frac{1}{4}}}\) can be written as:

\(\mathop {\lim }\limits_{x \to \frac{1}{4}} \frac{{\frac{1}{x} - 4}}{{x - \frac{1}{4}}} = \mathop {\lim }\limits_{x \to \frac{1}{4}} \frac{{\frac{1}{x} - \frac{1}{{\left( {1/4} \right)}}}}{{x - \frac{1}{4}}}\)

Find the value of a and function f

By the limit\(\mathop {\lim }\limits_{x \to \frac{1}{4}} \frac{{\frac{1}{x} - 4}}{{x - \frac{1}{4}}} = \mathop {\lim }\limits_{x \to \frac{1}{4}} \frac{{\frac{1}{x} - \frac{1}{{\left( {1/4} \right)}}}}{{x - \frac{1}{4}}}\), the value of a is \(\frac{1}{4}\).

The functionf can be expressed as:

\(f\left( x \right) = \frac{1}{x}\)

Thus, the function is \(f\left( x \right) = \frac{1}{x}\), and the value of a is \(\frac{1}{4}\).