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

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

Verified step by step guidance

1. 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 \(y = 0\).
2. 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.
3. 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.
4. Therefore, you can complete the statement: As \(x \to -\infty\), \(f(x) \to 0\).
5. This means the function values get closer and closer to zero but do not necessarily equal zero for very large negative \(x\).

Key Concepts

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.