How do you find the limit using the epsilon delta definition?
Proof
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To find the limit using the epsilon-delta definition, follow these steps:
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Understand the epsilon-delta definition: The limit of a function f(x) as x approaches a point c is L if for any positive value of epsilon (ε), there exists a positive value of delta (δ) such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
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Start by assuming the limit is L.
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Write the epsilon-delta definition: |f(x) - L| < ε.
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Manipulate the inequality to isolate f(x): -ε < f(x) - L < ε.
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Determine the conditions for which the inequality holds true: -ε + L < f(x) < ε + L.
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Express the condition in terms of the distance between x and c: 0 < |x - c| < δ.
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Combine the conditions: -ε + L < f(x) < ε + L and 0 < |x - c| < δ.
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Use algebraic manipulation to find a suitable expression for f(x) in terms of x and c.
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Determine the appropriate value of δ that satisfies the conditions.
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Prove that the chosen value of δ satisfies the conditions by showing that |f(x) - L| < ε whenever 0 < |x - c| < δ.
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Conclude that the limit of f(x) as x approaches c is L.
Note: The specific steps and calculations may vary depending on the function and the given values.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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