Question

Explain why the limits $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$and $\underset{z\to x}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}$are each initially in the form $\frac{0}{0}$. Why would cancelling a common factor of $h$or $z-x$be likely to resolve this indeterminate form?

Short answer

The value of $\frac{0}{0}$can't be defined or it can be said that its value doesn't exist. Therefore to define a limit at that particular point the factors should be cancelled from both numerator and denominators.

Step-by-step solution

Step 1. Given information

We have to say why these $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$and $\underset{z\to x}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}$are each initially in the form $\frac{0}{0}$and why would cancelling a common factor of $h$or $z-x$be likely to resolve this indeterminate form.

Step 2. Reason for the given question

The value of $\frac{0}{0}$can't be defined or it can be said that its value doesn't exist. Therefore to define a limit at that particular point the factors should be cancelled from both numerator and denominators.