Question

1. a table of selected values is given for a rational function ( p ).

( x )	( 4.9 )	( 4.99 )	( 4.999 )	( 5 )	( 5.001 )	( 5.01 )	( 5.1 )

a. estimate ( lim _{x
ightarrow 5^{-}} p(x) ) or explain why it does not exist.

b. estimate ( lim _{x
ightarrow 5^{+}} p(x) ) or explain why it does not exist.

c. does the graph of ( p ) have a hole, a vertical asymptote, or neither at ( x = 5 )? how do you know?

Explanation:

Step1: Analyze the left - hand limit

As \(x\) approaches \(5\) from the left (\(x\to5^{-}\)), we look at the values of \(p(x)\) for \(x = 4.9\), \(x = 4.99\), \(x=4.999\).
We observe that as \(x\) gets closer and closer to \(5\) from the left, \(p(x)\) is getting more and more negative.
We can approximate \(\lim_{x\to5^{-}}p(x)=-\infty\)

Step2: Analyze the right - hand limit

As \(x\) approaches \(5\) from the right (\(x\to5^{+}\)), we look at the values of \(p(x)\) for \(x = 5.001\), \(x = 5.01\), \(x = 5.1\).
We observe that as \(x\) gets closer and closer to \(5\) from the right, \(p(x)\) is getting more and more positive.
We can approximate \(\lim_{x\to5^{+}}p(x)=\infty\)

Step3: Determine the nature of the discontinuity

For a hole in the graph of a rational function \(y = p(x)=\frac{f(x)}{g(x)}\), \(\lim_{x\to a}p(x)\) is a finite number.
For a vertical asymptote at \(x = a\), \(\lim_{x\to a^{-}}p(x)=\pm\infty\) and \(\lim_{x\to a^{+}}p(x)=\pm\infty\)

Since \(\lim_{x\to5^{-}}p(x)=-\infty\) and \(\lim_{x\to5^{+}}p(x)=\infty\), the graph of \(p\) has a vertical asymptote at \(x = 5\)

Answer:

a. \(\lim_{x\to5^{-}}p(x)=-\infty\)

b. \(\lim_{x\to5^{+}}p(x)=\infty\)

c. The graph of \(p\) has a vertical asymptote at \(x = 5\) because \(\lim_{x\to5^{-}}p(x)=-\infty\) and \(\lim_{x\to5^{+}}p(x)=\infty\)