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



As $x\to -3^{-}$, $f\left(x\right)\to$____

Verified step by step guidance

1. Identify the point of interest on the x-axis, which is \(x \to -3^-\), meaning we are approaching \(-3\) from the left side.
2. Look at the graph near \(x = -3\) on the left side to observe the behavior of the function \(f(x)\) as \(x\) approaches \(-3\) from values less than \(-3\).
3. Notice the value of \(f(x)\) as \(x\) gets closer to \(-3\) from the left. Check if the function values increase without bound (go to \(+\infty\)), decrease without bound (go to \(-\infty\)), or approach a finite number.
4. From the graph, observe that as \(x\) approaches \(-3\) from the left, the function \(f(x)\) decreases without bound, meaning \(f(x) \to -\infty\).
5. Summarize the behavior: As \(x \to -3^-\), \(f(x) \to -\infty\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. They indicate 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 represents the value that the function approaches as x tends to positive or negative infinity. It shows the end behavior of the function. In this graph, the horizontal asymptote is y = 0, meaning the function values get closer to zero as x becomes very large or very small.

Limit Behavior Near a Point

The limit of a function as x approaches a specific value from the left or right describes the function's behavior near that point. For example, as x approaches -3 from the left (x → -3⁻), the function value approaches a certain number or infinity. Understanding this helps in interpreting the graph and completing limit statements.