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

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

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Identify the point of interest on the x-axis, which is $x \to -3^-$, meaning we are approaching $-3$ from the left side.

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 $-3$.

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.

From the graph, observe that as $x$ approaches $-3$ from the left, the function $f(x)$ decreases without bound, meaning $f(x) \to -\infty$.

Summarize the behavior: As $x \to -3^-$, $f(x) \to -\infty$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____