Answer
See the explanation
Work Step by Step
$r(x)=\frac{3x-10}{(x-2)^2}$,
a.
Table 1
$\begin{array}{II}
x & r(x)\\
1.5 & -22\\
1.9 & -430\\
1.99 & -40300\\
1.999 & -4003000
\end{array}$
Table 2
$\begin{array}{II}
x & r(x)\\
2.5 & -10\\
2.1 & -370\\
2.01 & -39700\\
2.001 & -3997000
\end{array}$
Table 3
$\begin{array}{II}
x & r(x)\\
10 & 0.3125\\
50 & 0.060764\\
100 & 0.030196\\
1000 & 0.00300199
\end{array}$
Table 4
$\begin{array}{II}
x & r(x)\\
-10 & -0.27777\\
-50 & -0.059172\\
-100 & -0.029796\\
-1000 & -0.002996
\end{array}$
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 $0^{+}$ from the right
- As $x$ approaches $-\infty$ (Table 4), $f(x)$ approaches $0^{-}$ from the left
The horizontal asymptote is $y=0$.
