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

As $x \to - \infty, f (x) \to_{_{}}$ _____

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Identify the horizontal asymptote from the graph. The horizontal asymptote is the line that the function approaches as $x$ goes to positive or negative infinity. In this graph, the horizontal asymptote is given as $y = 0$.

Recall that for rational functions, the behavior of $f(x)$ as $x \to -\infty$ (or $x \to \infty$) is determined by the horizontal asymptote, if one exists.

Since the horizontal asymptote is $y = 0$, this means that as $x$ becomes very large in the negative direction, the function values $f(x)$ approach 0.

Therefore, you can complete the statement: As $x \to -\infty$, $f(x) \to 0$.

This means the function values get closer and closer to zero but do not necessarily equal zero for very large negative $x$.

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Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. They represent values of x where the function is undefined, often due to division by zero in rational functions. In the graph, vertical asymptotes are shown at x = 6 and x = 14.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches positive or negative infinity. It indicates the value that the function approaches but does not necessarily reach. In this graph, the horizontal asymptote is y = 0, meaning as x goes to ±∞, f(x) approaches 0.

End Behavior of Rational Functions

The end behavior of a rational function describes how the function behaves as x approaches infinity or negative infinity. It is often determined by the degrees of the numerator and denominator polynomials. Here, since the horizontal asymptote is y = 0, as x → -∞, f(x) → 0.

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.As x→−∞, f(x)→x\(\to\)-\(\infty\),\(\text{ }\)f(x)\(\to\)_{_{}}_____