The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→1^+ f(x)

Verified step by step guidance

1. Identify that the limit \( \lim_{x \to 1^+} f(x) \) involves approaching the point \( x = 1 \) from the right.
2. Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1 from the right, \( f(x) \) will tend towards either positive or negative infinity.
3. Examine the behavior of \( f(x) \) as \( x \to 1^+ \) by considering the values of \( f(x) \) for \( x \) slightly greater than 1.
4. Determine whether \( f(x) \) increases without bound (approaches \( +\infty \)) or decreases without bound (approaches \( -\infty \)) as \( x \to 1^+ \).
5. Conclude the analysis of the limit based on the direction in which \( f(x) \) tends as \( x \to 1^+ \).

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 f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specific value from one side only. The notation lim x→1^+ f(x) indicates the limit of f(x) as x approaches 1 from the right (values greater than 1). Understanding one-sided limits is crucial for analyzing functions with vertical asymptotes.

Limit Behavior Near Asymptotes

The limit behavior near vertical asymptotes is characterized by the function's tendency to increase or decrease without bound. For instance, if lim x→1^+ f(x) approaches positive infinity, it indicates that as x gets closer to 1 from the right, the function's values rise indefinitely. This behavior is essential for understanding the overall shape and characteristics of the graph.