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

검증된 단계별 안내

Step 1: Understand the concept of a vertical asymptote. A vertical asymptote at x = a means that as x approaches a, the function h(x) tends to infinity or negative infinity.

Step 2: Identify the direction of approach. The limit x → -2^+ indicates that we are approaching x = -2 from the right side (values greater than -2).

Step 3: Analyze the behavior of h(x) as x approaches -2 from the right. Since there is a vertical asymptote at x = -2, observe whether h(x) increases towards positive infinity or decreases towards negative infinity.

Step 4: Use the graph to determine the behavior. Look at the graph of h(x) near x = -2 from the right side to see if the function is going upwards or downwards.

Step 5: Conclude the limit based on the observed behavior. If h(x) goes to positive infinity, the limit is positive infinity. If it goes to negative infinity, the limit is negative infinity.

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주요 개념

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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.

One-Sided Limits

One-sided limits evaluate the behavior of a function as the input approaches a specific value from one side only. The notation lim x→−2^+ h(x) indicates that we are interested in the limit of h(x) as x approaches -2 from the right (values greater than -2), which helps in understanding the function's behavior near the asymptote.

Limit Behavior Near Asymptotes

The behavior of limits near vertical asymptotes is crucial for understanding how functions behave at points where they are undefined. As x approaches a vertical asymptote, the function typically tends to either positive or negative infinity, which can be determined by analyzing the function's values just before and after the asymptote.

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