Explain the meaning of lim r(x) = L as x approaches b from the right.
Understand the Problem
The question is asking for an explanation of a mathematical limit expression. It describes the behavior of a function r(x) as x approaches a specific point b from the right, and how the function values behave in relation to a limit L.
Answer
As $ x $ approaches $ b $ from the right, the values of $ r(x) $ approach $ L $.
Answer for screen readers
The limit $ \lim_{x \to b^+} r(x) = L $ means that as $ x $ approaches the value $ b $ from the right, the values of the function $ r(x) $ approach the value $ L $.
Steps to Solve
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Understanding Limits from the Right In the expression $ \lim_{x \to b^+} r(x) = L $, we are focusing on the behavior of the function $ r(x) $ as $ x $ approaches the value $ b $ from values greater than $ b $. The $ + $ symbol indicates that we are only considering values of $ x $ that are slightly greater than $ b $.
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Defining the Limit L The limit $ L $ represents the value that the function $ r(x) $ will approach as $ x $ gets closer to $ b $ from the right side. In a mathematical sense, this means that for every value that is very close to $ L $, there is a corresponding interval around $ b $ (from the right) for which the values of $ r(x) $ get arbitrarily close to $ L $.
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Implication of the Limit This expression indicates that as we take values of $ x $ that are increasingly close to $ b $ (but always greater than $ b $), the values of $ r(x) $ will get closer to $ L $. In other words, for any small distance $\epsilon$ away from $ L $, there exists a small distance $\delta$ such that if $ 0 < x - b < \delta $, then $ |r(x) - L| < \epsilon $.
The limit $ \lim_{x \to b^+} r(x) = L $ means that as $ x $ approaches the value $ b $ from the right, the values of the function $ r(x) $ approach the value $ L $.
More Information
This concept is foundational in calculus and helps in understanding the behavior of functions at points where they may not be explicitly defined. The notation $ b^+ $ emphasizes the direction from which $ x $ approaches $ b $, reflecting how limits work in understanding continuity and function behavior.
Tips
- Confusing one-sided limits: Always remember that $ b^+ $ indicates approaching from the right side only, unlike $ b^- $, which approaches from the left.
- Misinterpreting the value of ( L ): It's essential to know that ( L ) is not necessarily the value of ( r(b) ); rather, it’s the value that ( r(x) ) approaches as ( x ) approaches ( b ).
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