Limits and the epsilon-delta Definition (Algebraic Translation)

Key Concepts

• Goal of limits: understand the behavior of f(x) as x approaches a point c, regardless of the value of f(c) or whether f is defined at c. The limit is l if, when x gets arbitrarily close to c, f(x) gets arbitrarily close to l.
• Distances on the real line are measured with absolute value: distance between a and b is
  |a-b|.
• The epsilon-delta language formalizes this:
  • For a given c, l, and a function f, we say lim_{x→c} f(x) = l if for every
    epsilon > 0 there exists a delta > 0 such that:
    0 < |x-c| < \delta \quad\Longrightarrow\quad |f(x) - l| < \epsilon.
  • The delta may depend on epsilon (and the function), but not on x.
• The domain condition: f must be defined on some open interval around c (i.e., there exists a neighborhood of c where f is defined).
• The role of absolute values: they express the idea of distance. We translate distance requirements from y to x using the actual formula for f.
• The graph-based intuition (a “graph game”) can be translated into algebra using the distance statements: we ask, “How close does x have to be to c so that f(x) is within epsilon of l?”
• It is crucial to preserve the logical direction: from the assumption about x (the if part) to the conclusion about f(x) (the then part). Reversing this implication is not generally valid without justification.
• Common approach in practice: for a given epsilon, solve the inequality |f(x) - l| < epsilon for x in terms of |x - c|, and identify a suitable delta that makes the implication work for all x in the neighborhood.

From Graph Game to Algebra

• Start with a concrete example to see how the graph idea translates to algebraic inequalities.
• Example setup: want to be within 2 units of l when l = 7, for a function y = f(x) with c = 4 (so we study lim_{x→4} f(x) = 7).
• Express the condition in terms of y and l:
  |y - 7| < 2.
• Replace y by its expression in x (here, y = 2x - 1):
  |2x - 1 - 7| < 2 \Rightarrow |2x - 8| < 2.
• Use properties of absolute value: |ab| = |a||b|, and |2| = 2, so
  |2x - 8| = 2|x - 4|. If we require 2|x-4| < 2, we get
  |x - 4| < 1. This tells us: being within 1 unit of x = 4 guarantees y is within 2 units of l = 7.
• More generally, to force |f(x) - l| < epsilon, you manipulate the inequality to isolate a distance in x:
  • Start with |f(x) - l| < \epsilon. Substitute f(x) and simplify to an expression involving |x - c|.
  • Factor or bound constants if possible to extract a form like |x - c| < \delta(\epsilon).
• For the specific example with epsilon = 2, the derived delta is 1 (i.e., any x with |x-4| < 1 suffices).
• For a smaller epsilon, say epsilon = 0.01, the bound tightens: from |2x - 8| < 0.01 we get 2|x-4| < 0.01 \Rightarrow |x-4| < 0.005. So delta = 0.005 works for epsilon = 0.01.
• Important idea: you can derive a general delta as a function of epsilon (here, delta = \epsilon/2 for this particular linear example), illustrating the dependence of delta on epsilon.

Worked Example 1: f(x) = 2x - 1, c = 4, l = 7

• Target: ensure |f(x) - l| < \epsilon.
• Compute: |f(x) - l| = |(2x - 1) - 7| = |2x - 8| = 2|x - 4|.
• Implication: if 0 < |x - 4| < \delta then |f(x) - 7| < \epsilon.
• To guarantee the inequality, set
  2|x - 4| < \epsilon \quad\Longrightarrow\quad |x - 4| < \frac{\epsilon}{2}.
• Therefore, one valid choice is \delta = \frac{\epsilon}{2}.
• Concrete instances:
  • If (\epsilon = 2): (\delta = 1). Hence, for all x with 0 < |x - 4| < 1, we have |f(x) - 7| < 2.
  • If (\epsilon = 0.01): (\delta = 0.005). Hence, for all x with 0 < |x - 4| < 0.005, we have |f(x) - 7| < 0.01.
• Observations:
  • This demonstrates turning a graph-like intuition into a precise algebraic bound.
  • The distance in y translates into a corresponding distance in x, scaled by the slope when f is linear.
  • The process shows how the definition requires the inequality to hold for all positive epsilon, not just one chosen value.

Worked Example 2: General linear case with a direct delta in terms of epsilon

• Function: f(x) = 5x - 3, c = 1, l = 2.
• Compute: |f(x) - l| = |(5x - 3) - 2| = |5x - 5| = 5|x - 1|.
• To guarantee |f(x) - l| < \epsilon, require
  5|x - 1| < \epsilon \quad\Longrightarrow\quad |x - 1| < \frac{\epsilon}{5}.
• Hence, a valid delta is \delta = \frac{\epsilon}{5}.
• Proof outline (standard structure):
  • For any (\epsilon > 0), choose (\delta = \epsilon/5).
  • If 0 < |x - 1| < \delta, then |f(x) - l| = 5|x - 1| < 5(\epsilon/5) = \epsilon.
• Important conceptual point: the delta can be a function of epsilon, and we must verify the implication holds for all x satisfying 0 < |x - c| < delta.
• Common pitfall discussed in the lecture: do not start with the “then” part (the |f(x) - l| < epsilon) and try to derive backwards from it without justification. The valid approach is to start from the left-hand side, bound it, and derive a corresponding delta that works for all x in the neighborhood.

The Formal Epsilon-Delta Definition (Clarity and Alignment)

• Setup: f is defined on an open interval about x = c.
• Statement: lim_{x→c} f(x) = l if and only if
  • For every epsilon > 0, there exists delta > 0 such that, for all x with 0 < |x - c| < delta,
    |f(x) - l| < \epsilon.
• Intuition alignment with the intuitive graph idea:
  • Given any desired closeness epsilon in the output (y-values), there is a corresponding closeness delta in the input (x-values) so that the output stays within epsilon for all inputs sufficiently close to c.
  • The delta may depend on epsilon and on the function, but not on the particular x chosen.
• Demonstration approach used in the lecture:
  • Start from |f(x) - l| < \epsilon, express in terms of x, and try to isolate a bound of the form |x - c| < delta(\epsilon).
  • Use properties of absolute values (e.g., |ab| = |a||b|) to separate factors when appropriate.
  • If the transformation seems to start from the “then” side, switch to the “if” side: assume |x - c| < delta and show |f(x) - l| < \epsilon for that delta.
• Example translation (short blind translation to illustrate the idea):
  • Take c = 4, f(x) = 2x - 1, l = 7.
  • The definition would read: For all (\epsilon > 0), there exists (\delta > 0) such that, if (0 < |x - 4| < \delta), then (|2x - 1 - 7| < \epsilon).
  • This aligns with the algebraic manipulations shown earlier, where you substitute and solve for delta in terms of epsilon.

Practical Takeaways and Strategy

• How to approach a limit problem in this framework:
  1) Identify c, l, and the formula for f(x).
  2) Write the core inequality |f(x) - l| < \epsilon and substitute f(x). 3) Use algebraic manipulations and absolute value rules to isolate a bound on |x - c|, aiming for a form like |x - c| < delta(\epsilon). 4) Choose delta as a simple, explicit function of epsilon (often delta = some constant * epsilon, depending on the function). 5) State the final delta-epsilon claim: For every (\epsilon > 0), there exists (\delta > 0) (e.g., delta = … ) such that 0 < |x - c| < delta implies |f(x) - l| < \epsilon.
• Important nuance: the existence of delta must be shown for all epsilon > 0, not just for one fixed epsilon. This is the heart of the formal definition.
• In linear examples, delta often scales with epsilon and the slope of the function (e.g