Answer
See the explanation
Work Step by Step
$r(x)=\frac{x}{x-2}$,
a.
Table 1
$\begin{array}{II}
x & r(x)\\
1.5 & -3\\
1.9 & -19\\
1.99 & -199\\
1.999 & -1999
\end{array}$
Table 2
$\begin{array}{II}
x & r(x)\\
2.5 & 5\\
2.1 & 21\\
2.01 & 201\\
2.001 & 2001
\end{array}$
Table 3
$\begin{array}{II}
x & r(x)\\
10 & 1.25\\
50 & 1.04167\\
100 & 1.02041\\
1000 & 1.00200401
\end{array}$
Table 4
$\begin{array}{II}
x & r(x)\\
-10 & 0.833\\
-50 & 0.96154\\
-100 & 0.9803922\\
-1000 & 0.998004
\end{array}$
b.
- As $x$ approaches $2^{-}$ from the left(Table 1), $f(x)$ approaches $-\infty$
- As $x$ approaches $2^{+}$ from the right(Table 2), $f(x)$ approaches $\infty$
c.
- As $x$ approaches $\infty$ (Table 3), $f(x)$ approaches $1^{+}$ from the right
- As $x$ approaches $-\infty$ (Table 4), $f(x)$ approaches $1^{-}$ from the left
The horizontal asymptote is $y=1$.
