Question

student activity sheet 5; exploring \asymptotes and removable discontinuities\ page 2 of 11

1. consider the quotient form of ( r(x) ). for large values of ( x ), which terms have the greatest impact on the function values? which terms have very little impact?
2. the parent rational function ( y=\frac{1}{x} ) is undefined at ( x = 0 ). complete the table to evaluate this function for values of ( x ) very near 0. what do the values in the table indicate about the behavior of the graph?

( x )

( y=\frac{1}{x} )

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-1

-1

-0.1

-10

-0.01

-100

-0.001

-1000

0

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0.001

1000

0.01

100

0.1

10

1

1

Explanation:

Step1: Analyze the function values

For \(x = - 1\), \(y=\frac{1}{-1}=-1\); for \(x=-0.1\), \(y = \frac{1}{-0.1}=-10\); for \(x=-0.01\), \(y=\frac{1}{-0.01}=-100\); for \(x=-0.001\), \(y=\frac{1}{-0.001}=-1000\); for \(x = 0.001\), \(y=\frac{1}{0.001}=1000\); for \(x=0.01\), \(y=\frac{1}{0.01}=100\); for \(x = 0.1\), \(y=\frac{1}{0.1}=10\); for \(x = 1\), \(y=\frac{1}{1}=1\).

Step2: Determine the behavior

As \(x\to0^{-}\) (approaching \(0\) from the left, \(x<0\)), \(y\to-\infty\). As \(x\to0^{+}\) (approaching \(0\) from the right, \(x > 0\)), \(y\to+\infty\).

Answer:

The values in the table indicate that the line \(x = 0\) is a vertical asymptote of the graph of \(y=\frac{1}{x}\). As \(x\) approaches \(0\) from the left (\(x<0,x\to0\)), \(y\) approaches \(-\infty\). As \(x\) approaches \(0\) from the right (\(x>0,x\to0\)), \(y\) approaches \(+\infty\).