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

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

Verified step by step guidance

1. Identify the point of interest on the x-axis, which is \(x \to 1^-\), meaning we are approaching 1 from the left side.
2. 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.
3. Notice that the graph is very close to the horizontal asymptote \(y = 0\) near \(x = 1\), and the function values are slightly below zero.
4. 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 \(x \to 1^-\).
5. Therefore, the limit of \(f(x)\) as \(x\) approaches 1 from the left is 0, but the function values are slightly negative.

Key Concepts

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.