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)

Guida verificata passo dopo passo

Step 1: Identify the type of limit problem. Since the limit is approaching a point where there is a vertical asymptote (x = 3), this is a limit involving a vertical asymptote.

Step 2: Understand the behavior of the function near the asymptote. As x approaches 3, the function h(x) will tend to either positive infinity or negative infinity, depending on the direction from which x approaches 3.

Step 3: Consider the one-sided limits. Evaluate the limit from the left (x → 3⁻) and from the right (x → 3⁺) separately to determine the behavior of h(x) as it approaches the asymptote.

Step 4: Analyze the graph. Look at the graph of h(x) near x = 3 to see if the function approaches positive or negative infinity from either side.

Step 5: Conclude the limit. Based on the behavior observed in the graph, determine if the two one-sided limits are equal or if they differ, which will indicate whether the overall limit exists or not.

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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, h(x) will not approach a finite limit.

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Introduction to Cotangent Graph Example 1

Limits

A limit describes the behavior of a function as the input approaches a particular value. In the context of the question, evaluating the limit as x approaches 3 for h(x) involves determining how h(x) behaves near the vertical asymptote at x = 3, which typically results in the limit being either positive or negative infinity.

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One-Sided Limits

One-sided limits are used to analyze the behavior of a function as it approaches a specific point from one side only, either from the left (denoted as lim x→c-) or from the right (denoted as lim x→c+). For the limit as x approaches 3, it is essential to consider both one-sided limits to fully understand the behavior of h(x) near the vertical asymptote.

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One-Sided Limits

Pratica correlata

Domanda del libro di testo

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h$\left$(2$\right$)$