Answer
See the explanation
Work Step by Step
$r(x)=\frac{3x^2+1}{(x-2)^2}$,
a.
Table 1
$\begin{array}{II}
x & r(x)\\
1.5 & 31\\
1.9 & 1183\\
1.99 & 128803\\
1.999 & 12988003
\end{array}$
Table 2
$\begin{array}{II}
x & r(x)\\
2.5 & 79\\
2.1 & 1423\\
2.01 & 131203\\
2.001 & 13012003
\end{array}$
Table 3
$\begin{array}{II}
x & r(x)\\
10 & 4.703125\\
50 & 3.2556423\\
100 & 3.1238025\\
1000 & 3.0120371
\end{array}$
Table 4
$\begin{array}{II}
x & r(x)\\
-10 & 2.0902777\\
-50 & 2.7740384\\
-100 & 2.883602\\
-1000 & 2.9880369
\end{array}$
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 $3^{+}$ from the right
- As $x$ approaches $-\infty$ (Table 4), $f(x)$ approaches $3^{-}$ from the left
The horizontal asymptote is $y=3$.
