It looks like you're trying to analyze the problem of finding a suitable \(\delta\) in order to satisfy the condition for a limit, specifically for the function \(f(x) = 2x - 5\) as \(x\) approaches \(6\). The goal is to ensure that \(|f(x) - 7| < \epsilon\) whenever \(|x - 6| < \delta\).
Let's break it down step-by-step:
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Determine the expression:
We want to analyze \(|f(x) - 7|\):
\[ f(x) = 2x - 5 \] Therefore,
\[ |f(x) - 7| = |(2x - 5) - 7| = |2x - 12| = 2|x - 6| \]
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Set the goal:
We want this expression to be less than \(\epsilon\):
\[ |2x - 12| < \epsilon \]
From our earlier calculation:
\[ 2|x - 6| < \epsilon \]
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Manipulate the inequality:
To make this easier, we can divide both sides by \(2\):
\[ |x - 6| < \frac{\epsilon}{2} \]
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Choose \(\delta\):
Thus, we can choose:
\[ \delta = \frac{\epsilon}{2} \]
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Conclusion:
In summary, if we choose \(\delta = \frac{\epsilon}{2}\), then whenever \(0 < |x - 6| < \delta\), it will guarantee that:
\[ |(2x - 5) - 7| < \epsilon \]
This procedure is common in proving limits. The approach you've taken is a good way to analyze the continuity of the function at the given point. If you have additional questions or need clarification on this topic, feel free to ask!