Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As $x\to 1^{+},f\left(x\right)\to$ __

Verified step by step guidance

1. Identify the vertical asymptote near the value x = 1 on the graph. Notice that the graph has vertical asymptotes at x = -16 and x = -8, but none at x = 1, so the behavior near x = 1 is not influenced by a vertical asymptote.
2. Observe the behavior of the function f(x) as x approaches 1 from the right (x \to 1^+). Look at the graph closely near x = 1 and see whether the function values increase, decrease, or approach a specific number.
3. From the graph, as x approaches 1 from the right, the function values seem to approach a certain y-value. Determine if the function is going towards positive infinity, negative infinity, or a finite number.
4. Since the graph near x = 1 shows the function values approaching a finite number, identify that number by looking at the y-axis value the graph approaches.
5. Conclude that as x \to 1^+, f(x) approaches the finite value you identified from the graph.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as x approaches a specific value. They represent values of x that make the denominator zero in a rational function, causing the function to be undefined. The graph shows the function's behavior near these lines, indicating limits from the left or right.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. They indicate the value that the function approaches but does not necessarily reach. For rational functions, horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials.

Limit Behavior Near Asymptotes

The limit of a function as x approaches a certain value from the right or left describes how the function behaves near that point. Near vertical asymptotes, the function may approach positive or negative infinity. Understanding this helps in completing statements about the function's behavior as x approaches specific values.