The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→3^+ h(x)

Verified step by step guidance

1. Identify the behavior of the function h(x) as x approaches 3 from the right (x → 3^+).
2. Recognize that a vertical asymptote at x = 3 indicates that the function h(x) becomes unbounded as x approaches 3.
3. Since we are considering the limit from the right (x → 3^+), observe the direction in which h(x) tends as x gets closer to 3 from values greater than 3.
4. Determine whether h(x) approaches positive infinity or negative infinity as x approaches 3 from the right, based on the graph's behavior near the asymptote.
5. Conclude the limit by stating that lim x→3^+ h(x) is either positive infinity or negative infinity, depending on the observed behavior.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function h has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, the function's output becomes unbounded.

Limits

A limit describes the behavior of a function as the input approaches a particular value. The notation lim x→3^+ h(x) specifically refers to the limit of h(x) as x approaches 3 from the right side, which is crucial for understanding how the function behaves near the vertical asymptote at x = 3.

One-Sided Limits

One-sided limits evaluate the behavior of a function as the input approaches a specific point from one direction only. The notation lim x→3^+ h(x) indicates a right-hand limit, which helps determine the value that h(x) approaches as x gets closer to 3 from values greater than 3, providing insight into the function's behavior near the asymptote.