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Magazine Chapter 3: Problem 26
In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 2^{+}} \frac{2}{x^{2}-4} $$
Short Answer
Expert verified
The limit is positive infinity, \( \lim_{x \to 2^+} \frac{2}{x^2 - 4} = +\infty \).
Step by step solution
01
Understand the Limit Expression
The expression given is \( \lim_{x \to 2^+} \frac{2}{x^2 - 4} \). This means we need to find the limit of the function \( \frac{2}{x^2 - 4} \) as \( x \) approaches 2 from the right (from values greater than 2).
02
Rewrite the Denominator
Notice that the denominator \( x^2 - 4 \) can be rewritten as \( (x-2)(x+2) \). This factorization will help us understand what's happening around the point \( x = 2 \).
03
Consider Values Close to 2 from the Right
Examine the behavior of the function as \( x \) approaches 2 from the right (e.g., 2.1, 2.01, 2.001). For each of these values, substitute into the denominator: \( (x-2) \) is positive but very small, and \( (x+2) \) is slightly greater than 4, thus giving a very small positive denominator.
04
Analyze the Function's Behavior
Substitute these values into the function \( \frac{2}{x^2 - 4} = \frac{2}{(x-2)(x+2)} \). As \( x \) approaches 2 from the right, \( (x-2) \rightarrow 0^+ \) which makes the entire expression approach \( +\infty \). Therefore, the function approaches positive infinity.
05
Conclusion from Table/Graph Exploration
Using a table or a graph can confirm these calculations. If you plot the graph of \( \frac{2}{x^2 - 4} \) or create a table of values for \( x \) slightly greater than 2, you will observe the values increasingly grow larger, indicating a trend towards positive infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Investigation
When investigating limits in calculus, we want to understand the behavior of a function as the input values get very close to a specific number. In the exercise, the limit expression is \( \lim_{x \to 2^+} \frac{2}{x^2 - 4} \). This tells us to look at the behavior of the function \( \frac{2}{x^2 - 4} \) as \( x \) approaches 2 but only from the right side. To do this effectively, we can make use of tables and graphs.
- Tables: Create a list of \( x \) values that are a little more than 2, like 2.1, 2.01, and 2.001, and calculate the corresponding function values. This shows how the function behaves as it gets close to that \( x \, \) value.
- Graphs: Sketch or use software to plot the function near \( x = 2 \). Notice how the function behaves as it gets close to 2, confirming our calculations and tables.
One-Sided Limits
One-sided limits focus on the function's behavior from one specific direction. For example, \( \lim_{x \to 2^+} \frac{2}{x^2 - 4} \) is a limit approaching 2 from the right, which is denoted as \( 2^+ \).
- In one-sided limits, the function's behavior could be different when approaching from either direction, so it's often necessary to explicitly state which side you're analyzing.
- In our problem, approaching 2 from the right means looking at \( x \) values slightly larger than 2, like 2.1 or 2.01. This directionality can greatly affect the behavior of the function; hence, it's crucial to specify and analyze the side pertinently.
Function Behavior Analysis
Function behavior analysis in limits is about understanding how functions act near a particular point. We explore trends and values the function might approach. When analyzing \( \frac{2}{x^2 - 4} \) as \( x \to 2^+ \), it's essential to factor the denominator into \( (x-2)(x+2) \).
- At around \( x = 2 \), \( (x-2) \) becomes tiny but positive, while \( (x+2) \) remains slightly above 4. This means the denominator becomes a small positive number, leading the whole function to grow larger, heading towards positive infinity.
- Seeing how small changes in \( x \) affect the function helps us predict this behavior. Analyzing these minimal changes is crucial to understanding how a function behaves near significant points like discontinuities.
Infinity in Calculus
Infinity has a special place in calculus, where it often describes limits and behaviors. In our example, as \( x \to 2^+ \), \( \frac{2}{x^2 - 4} \) approaches positive infinity. This means the function increases beyond all bounds:
- When analyzing, if the limit is infinity, it indicates the function's values grow very large without bound as \( x \) nears the specified number, not settling to a fixed number.
- Identifying whether a function approaches positive or negative infinity helps in understanding its vertical asymptotes and general behavior. For \( x > 2 \), the expression \( (x-2) \to 0^+ \) makes the overall value of the function trend towards \(+\infty\).
