How do you find the limit using the epsilon delta definition?

Answer 1
The #epsilon, delta#-definition can be used to formally prove a limit; however, it is not used to find the limit. Let us use the epsilon delta definition to prove the limit: #lim_{x to 2}(2x-3)=1#

Proof

For all #epsilon>0#, there exists #delta=epsilon/2>0# such that #0<|x-2| < deltaRightarrow |x-2| < epsilon/2 Rightarrow 2|x-2|< epsilon# #Rightarrow |2x-4| < epsilon Rightarrow|(2x-3)-1| < epsilon#
Hence, #lim_{x to 2}(2x-3)=1#.
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Answer 2

To find the limit using the epsilon-delta definition, follow these steps:

  1. 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.

  2. Start by assuming the limit is L.

  3. Write the epsilon-delta definition: |f(x) - L| < ε.

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

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

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

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

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

  9. Determine the appropriate value of δ that satisfies the conditions.

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

  11. 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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Answer from HIX Tutor

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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