Question

Zero and Divisions. In `f(x) = (27x-(3^x)x) / ((3^x)x-9x)` find the Zero, Undefined, and Indeterminate. Please teach me how?

Answers are:
Zero = 3
Undefined = 2
Indeterminate = 0
I just don't know how to get them.

Answer 1

For each of these, we need to think about where the zeroes for the numerator and denominator are. Let's explicitly write that

`n(x) = (27-3^x)x and d(x) = (3^x-9)x`

Clearly, `n(x) = 0` when `x = 0 or 3` and `d(x) = 0` when `x = 0, 2`.

A zero happens when `n(x)` is zero and `d(x)` is some number, so we get `0/text(some number) = 0`.

An undefined value happens whens `d(x)` is zero and `n(x)` is some number, so we get `text(some number)/0` which is undefined.

An indeterminate case is if both `d(x)` and `n(x)` are zero, since `0/0` is indeterminate.

Thinking about all three of these cases, we can easily derive the solution you gave: zero at `x=3`, undefined at `x=2` and indeterminate at `x=0`.