Question

student activity sheet 5; exploring \asymptotes and removable discontinuities\ page 3 of 11

1. try the same analysis for the function ( r(x)=\frac{3 x + 8}{x + 2} ). complete the table to evaluate this function for values of ( x ) very near -2. what do the values in the table indicate about the behavior of the graph?

( x )	( r(x)=\frac{3 x + 8}{x + 2} )
-2.1	-17
-2.01
-2.001
-2	undefined
-1.999
-1.99
-1.9
-1

Explanation:

Step1: Calculate \( r(-2.01) \)

Substitute \( x = - 2.01 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK0
$$

Step2: Calculate \( r(-2.001) \)

Substitute \( x=-2.001 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK1
$$

Step3: Calculate \( r(-1.999) \)

Substitute \( x = - 1.999 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK2
$$

Step4: Calculate \( r(-1.99) \)

Substitute \( x=-1.99 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK3
$$

Step5: Calculate \( r(-1.9) \)

Substitute \( x=-1.9 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK4
$$

Step6: Calculate \( r(-1) \)

Substitute \( x = - 1 \) into \( r(x)=\frac{3x + 8}{x + 2} \).

$$
LATEXBLOCK5
$$

As \( x\) approaches \(-2\) from the left (\(x=-2.1,x = - 2.01,x=-2.001\)), \(r(x)\) approaches \(-\infty\). As \(x\) approaches \(-2\) from the right (\(x=-1.999,x=-1.99,x=-1.9\)), \(r(x)\) approaches \(+\infty\). This indicates that \(x = - 2\) is a vertical asymptote of the function \(y = r(x)\).

Answer:

\(x\)	\(r(x)=\frac{3x + 8}{x + 2}\)
\(-2.1\)	\(-17\)
\(-2.01\)	\(-197\)
\(-2.001\)	\(-1997\)
\(-2\)	Undefined
\(-1.999\)	\(2003\)
\(-1.99\)	\(203\)
\(-1.9\)	\(23\)
\(-1\)	\(5\)

The values in the table indicate that \(x=-2\) is a vertical asymptote of the function \(r(x)=\frac{3x + 8}{x + 2}\). As \(x\) approaches \(-2\) from the left, \(r(x)\) approaches \(-\infty\), and as \(x\) approaches \(-2\) from the right, \(r(x)\) approaches \(+\infty\).