Epsilon-Delta Definition of Limits

Epsilon-Delta Definition of Limits

• The epsilon-delta definition is a formal way to express the idea of a limit in mathematics, providing a precise way to talk about how close the output of a function gets to a specific value as the input approaches another specific value.

Key Components of Epsilon-Delta Definition

• ε (epsilon) represents a positive real number signifying a small tolerance level, where the function's output should be within this tolerance level of the limit value.
• δ (delta) represents another positive real number, signifying how close the input of the function needs to be to the limit point to guarantee the output falls within the tolerance level (ε).

Epsilon-Delta Definition Statement

• For any function f(x) and a number L, the limit of f(x) as x approaches a (lim_(x->a) f(x) = L) exists if for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, then |f(x) - L| < ε.

Simplified Explanation of Epsilon-Delta Definition

• No matter how small a positive tolerance level (ε) you choose, there exists a positive distance (δ) around the limit point (a), such that whenever the input (x) falls within that distance of the limit point, the function's output (f(x)) is guaranteed to be within the chosen tolerance level (ε) of the limit value (L).

Proving Limits with Epsilon-Delta

• To prove a limit with epsilon-delta, follow these steps:
  • Start with the definition, acknowledging that the proof needs to hold true for any chosen positive value of epsilon.
  • Find a suitable value of delta (δ) that works for the specific function and limit you're proving by manipulating the function's formula and inequalities.
  • Demonstrate that for the chosen delta (δ), whenever the distance from the input to the limit point is less than delta, the distance from the function's output to the limit value is indeed less than the chosen epsilon.

Example of Proving Limits with Epsilon-Delta

• Prove that the limit of f(x) = x^2 as x approaches 2 is equal to 4 (lim_(x->2) (x^2) = 4) by:
  • Starting with the definition, "For any ε > 0..."
  • Finding a suitable delta (δ) by manipulating the inequality (x - 2)^2 < ε.
  • Showing that for 0 < |x - 2| < δ, then |(x^2) - 4| = |(x - 2)(x + 2)| < ε, which holds true because both (x - 2) and (x + 2) are within δ of 2 and their product is less than ε due to the chosen delta.

Key Points

• The epsilon-delta definition provides a rigorous way to express the concept of limits.
• Proving limits with epsilon-delta involves finding a suitable delta for a chosen epsilon to ensure the function's output falls within the tolerance level of the limit value.
• This method requires manipulating inequalities and the function's formula.