Answer
See the explanation
Work Step by Step
$r(x)=\frac{4x+1}{x-2}$,
a.
Table 1
$\begin{array}{II}
x & r(x)\\
1.5 & -14\\
1.9 & -86\\
1.99 & -896\\
1.999 & -8996
\end{array}$
Table 2
$\begin{array}{II}
x & r(x)\\
2.5 & 22\\
2.1 & 94\\
2.01 & 904\\
2.001 & 9004
\end{array}$
Table 3
$\begin{array}{II}
x & r(x)\\
10 & 5.125\\
50 & 4.1875\\
100 & 4.0918\\
1000 & 4.009018
\end{array}$
Table 4
$\begin{array}{II}
x & r(x)\\
-10 & 3.25\\
-50 & 3.8269\\
-100 & 3.91176\\
-1000 & 3.99101796
\end{array}$
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 $4^{+}$ from the right
- As $x$ approaches $-\infty$ (Table 4), $f(x)$ approaches $4^{-}$ from the left
The horizontal asymptote is $y=4$.
