How do you find the limit using the epsilon delta definition?
Proof
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To find the limit using the epsilondelta definition, follow these steps:

Understand the epsilondelta 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.

Start by assuming the limit is L.

Write the epsilondelta definition: f(x)  L < ε.

Manipulate the inequality to isolate f(x): ε < f(x)  L < ε.

Determine the conditions for which the inequality holds true: ε + L < f(x) < ε + L.

Express the condition in terms of the distance between x and c: 0 < x  c < δ.

Combine the conditions: ε + L < f(x) < ε + L and 0 < x  c < δ.

Use algebraic manipulation to find a suitable expression for f(x) in terms of x and c.

Determine the appropriate value of δ that satisfies the conditions.

Prove that the chosen value of δ satisfies the conditions by showing that f(x)  L < ε whenever 0 < x  c < δ.

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 onesided 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 onesided 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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