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2-25.

Write a complete set of approach statements for $y = \frac { ( 3 x - 1 ) ( x + 2 ) } { 3 x - 1 }$.

A complete set of approach statements will include information about what happens when $x \rightarrow \infty$ and $x \rightarrow −\infty$. (This will tell us about the location of horizontal asymptotes, if any.) Also, a complete set of approach statements will describe what happens on each side of a hole or vertical asymptote. So the first step will be to determine the location of holes and vertical asymptotes.

$y = \frac{(3x-1)(x+2)}{(3x-1)}=x+2 \text{ with a hole at }x=\frac{1}{3} \text{ and no vertical asymptotes}$

Now that we have determined the location of holes and vertical asymptotes, we can write a complete set of approach statement:

$x \rightarrow −\infty, y \rightarrow$ __________
$x \rightarrow −2^−, y \rightarrow$ _________
$x \rightarrow \infty, y \rightarrow$ __________
$x \rightarrow −2^+, y \rightarrow$ ________

Use the simplified equation, $y = x + 2$, to evaluate these approach statements.