Chapter 3.2, Problem 5QR

To determine

Whether the limit could be used to define $f\text{'}\left(a\right)$ .

Answer to Problem 5QR

The limit cannot be used to define $f^{\prime }\left(a\right)$ .

Explanation of Solution

Given information:

The limit function $=\underset{x\to a}{\lim }\frac{f\left(a+h\right)-f\left(a-h\right)}{h}$

Calculation:

The given limit function $=\underset{x\to a}{\lim }\frac{f\left(a+h\right)-f\left(a-h\right)}{h}$

Since, the given function is not possible to find the derivative of $x$ at $f\left(a\right)$ .

Thus, the correct function is $f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a-h\right)}{2h}$

The quotient is found as shown in the graph (divide the $y$ distance by $x$ distance).

Then, calculate the derivative as the limit of that quotient as the two points combined.

  

Hence, The limit cannot be used to define $f^{\prime }\left(a\right)$ .