Question

use the graph of the rational function in the figure shown to complete each statement in exercises 9–14.

1. as ( x

ightarrow - 3^{-} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow - 3^{+} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow 1^{-} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow 1^{+} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow -infty ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow infty ), ( f(x)
ightarrow ) ______.
use the graph of the rational function in the figure shown to complete each statement in exercises 15–20.

1. as ( x

ightarrow 1^{+} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow 1^{-} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow - 2^{+} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow - 2^{-} ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow infty ), ( f(x)
ightarrow ) ______.

1. as ( x

ightarrow -infty ), ( f(x)
ightarrow ) ______.

Explanation:

Step1: Analyze $x\to-3^-$ (left of -3)

The graph goes to $-\infty$ as $x$ approaches -3 from the left.

Step2: Analyze $x\to-3^+$ (right of -3)

The graph goes to $+\infty$ as $x$ approaches -3 from the right.

Step3: Analyze $x\to1^-$ (left of 1)

The graph goes to $-\infty$ as $x$ approaches 1 from the left.

Step4: Analyze $x\to1^+$ (right of 1)

The graph goes to $+\infty$ as $x$ approaches 1 from the right.

Step5: Analyze $x\to-\infty$ (left end behavior)

The graph approaches the horizontal asymptote $y=0$.

Step6: Analyze $x\to\infty$ (right end behavior)

The graph approaches the horizontal asymptote $y=0$.

Step7: Analyze $x\to1^+$ (second graph, right of 1)

The graph goes to $+\infty$ as $x$ approaches 1 from the right.

Step8: Analyze $x\to1^-$ (second graph, left of 1)

The graph goes to $-\infty$ as $x$ approaches 1 from the left.

Step9: Analyze $x\to-2^+$ (second graph, right of -2)

The graph goes to $-\infty$ as $x$ approaches -2 from the right.

Step10: Analyze $x\to-2^-$ (second graph, left of -2)

The graph goes to $+\infty$ as $x$ approaches -2 from the left.

Step11: Analyze $x\to\infty$ (second graph, right end)

The graph approaches the horizontal asymptote $y=1$.

Step12: Analyze $x\to-\infty$ (second graph, left end)

The graph approaches the horizontal asymptote $y=1$.

Answer:

1. $-\infty$
2. $+\infty$
3. $-\infty$
4. $+\infty$
5. $0$
6. $0$
7. $+\infty$
8. $-\infty$
9. $-\infty$
10. $+\infty$
11. $1$
12. $1$