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

As $x \to 1^{-}, f (x) \to_{_{}}$ ______

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Identify the point of interest on the x-axis, which is $x \to 1^-$, meaning we are approaching 1 from the left side.

Look at the graph near $x = 1$ and observe the behavior of the function $f(x)$ as $x$ approaches 1 from values less than 1.

Notice that the graph is very close to the horizontal asymptote $y = 0$ near $x = 1$, and the function values are slightly below zero.

Since the function values are approaching zero from the negative side as $x$ approaches 1 from the left, we conclude that $f(x) \to 0^-$ as $x \to 1^-$.

Therefore, the limit of $f(x)$ as $x$ approaches 1 from the left is 0, but the function values are slightly negative.

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주요 개념

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

Vertical Asymptotes

Vertical asymptotes occur where a 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 a function approaches as x approaches 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 1 from the left (x → 1⁻), the function value approaches a certain number or infinity. Understanding this helps in analyzing function continuity and asymptotic behavior.

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.As x→1−, f(x)→x\(\to\)1^{-},\(\text{ }\)f(x)\(\to\)_{_{}}______