Question

ating a rational function
x | f(x)
-1 | -1
-0.1 | -10
-0.01 | -100
-0.001 | -1000
x | f(x)
0.001 | 1000
0.01 | 100
0.1 | 10
1 | 1
what happens as x approaches zero from the positive direction?
what happens as x approaches zero from the negative direction?
what is the domain of the function?

Explanation:

Step1: Analyze positive approach to 0

Looking at the right table (positive x values: 0.001, 0.01, 0.1, 1), as \( x \) decreases towards 0 (from 1 to 0.001), \( f(x) \) values are 1, 10, 100, 1000. So \( f(x) \) increases without bound (goes to \( +\infty \)).

Step2: Analyze negative approach to 0

Left table (negative x values: -1, -0.1, -0.01, -0.001), as \( x \) approaches 0 from the left (becomes less negative, closer to 0), \( f(x) \) values are -1, -10, -100, -1000. So \( f(x) \) decreases without bound (goes to \( -\infty \)).

Step3: Determine the domain

A rational function (implied here, likely \( f(x)=\frac{1}{x} \) from the table values: \( f(-1)=\frac{1}{-1}=-1 \), \( f(0.001)=\frac{1}{0.001}=1000 \), etc.) has a domain excluding \( x = 0 \) (since division by zero is undefined). So domain is all real numbers except \( x = 0 \), or \( \{x \in \mathbb{R} | x
eq 0\} \).

Answer:

s:

1. As \( x \) approaches zero from the positive direction, \( f(x) \) approaches \( +\infty \) (increases without bound).
2. As \( x \) approaches zero from the negative direction, \( f(x) \) approaches \( -\infty \) (decreases without bound).
3. The domain of the function is all real numbers except \( x = 0 \) (or \( \{x \in \mathbb{R} | x

eq 0\} \)).