Question

fill in the following table for values of x near 3. what do you observe about the value of f(x) as x approaches 3 from the right? from the left? f(x)=\frac{1}{x - 3}

x	2.9	2.99	2.999	3.001	3.01	3.1

as x approaches 3 from the left, f(x) approaches. select an answer
as x approaches 3 from the right, f(x) approaches. select an answer

question help: video ebook written example

Explanation:

Step1: Substitute x - values into function

For \(x = 2.9\), \(f(2.9)=\frac{1}{2.9 - 3}=\frac{1}{- 0.1}=-10\).
For \(x = 2.99\), \(f(2.99)=\frac{1}{2.99 - 3}=\frac{1}{-0.01}=-100\).
For \(x = 2.999\), \(f(2.999)=\frac{1}{2.999 - 3}=\frac{1}{-0.001}=-1000\).
For \(x = 3.001\), \(f(3.001)=\frac{1}{3.001 - 3}=\frac{1}{0.001}=1000\).
For \(x = 3.01\), \(f(3.01)=\frac{1}{3.01 - 3}=\frac{1}{0.01}=100\).
For \(x = 3.1\), \(f(3.1)=\frac{1}{3.1 - 3}=\frac{1}{0.1}=10\).

Answer:

\(x\)	\(f(x)\)
\(2.99\)	\(-100\)
\(2.999\)	\(-1000\)
\(3.001\)	\(1000\)
\(3.01\)	\(100\)
\(3.1\)	\(10\)

As \(x\) approaches \(3\) from the left (\(x<3\)), \(f(x)\) approaches \(-\infty\). As \(x\) approaches \(3\) from the right (\(x > 3\)), \(f(x)\) approaches \(\infty\).