Epsilon-Delta Definition of Limits: Solving Problems

Overview

• Introduction: Solving various limit problems using the epsilon-delta definition.
• Prerequisites: OGE video (one problem discussion) and theory video (epsilon-delta definition theory).
• Types of Problems: Linear, quadratic (two and three terms), cubic, square root functions, rational functions, square root in rational form.
• Total Problems: Nine problems covered.

Epsilon-Delta Definition

• Definition: The limit of f(x) as x approaches a is L if for every ε > 0, there exists a δ > 0 such that for all x in the domain of the function, if 0 < |x - a| < δ, then |f(x) - L| < ε.

Problem 1: Linear Function

• Problem: Prove that limit as x approaches 8 of (1/2)x - 3 is 1.
• Strategy: Use epsilon-delta definition.
• Steps:
  • Choose ε > 0 and find δ > 0 such that |x - 8| < δ implies |(1/2)x - 4| < ε.
  • Simplify |(1/2)x - 4| to |x - 8| < 2ε.
  • Choose δ = 2ε.
  • Show that 0 < |x - 8| < δ implies |(1/2)x - 4| < ε, thus completing the proof.

Problem 2: Cubic Function

• Problem: Prove that limit as x approaches 2 of x^3 is 8.
• Strategy: Factor the expression using difference of cubes.
• Steps:
  • Factor x^3 - 8 as (x - 2)(x^2 + 2x + 4).
  • Bound the remaining factor |x^2 + 2x + 4| when |x - 2| < 1, yielding |x^2 + 2x + 4| ≤ 19.
  • Choose δ such that 19δ ≤ ε, or δ ≤ ε/19.
  • Use minimum condition to select appropriate δ.
  • Show 19|x - 2| < ε hence complete the proof.

Problem 3: Square Root Function

• Problem: Prove that limit as t approaches 3 of sqrt(t + 1) is 2.
• Strategy: Manipulate algebraically and use conjugate.
• Steps:
  • Multiply by the conjugate to simplify the expression.
  • Bound the resultant factor and choose δ accordingly.
  • Ensure δ < ε/19 and use the minimum function to select δ.
  • Show that the appropriate bounds imply the desired inequality.

Problem 4: Quadratic Function

• Problem: Prove that limit as x approaches 2 of 3x^2 + 4 is 16.
• Strategy: Factor polynomial and bound remaining terms.
• Steps:
  • Simplify |3x^2 + 4 - 16| and factor further.
  • Bound the factor 3|x + 2| when |x - 2| < 1, yielding 3|x + 2| ≤ 15.
  • Choose δ such that 15δ ≤ ε, or δ ≤ ε/15.
  • Show that the condition holds, thus proving the limit.

Problem 5: Polynomial Function

• Problem: Prove that limit as x approaches 1 of 4 + x - 3x^2 is 2.
• Strategy: Simplify polynomial and solve.
• Steps:
  • Simplify |4 + x - 3x^2 - 2| through algebraic manipulation.
  • Factorize further and bound the appropriate term.
  • Choose δ considering the bounds and show the limit.

Problem 6: Rational Function

• Problem: Prove that limit as u approaches 1 of 3/(u + 2) is 1.
• Strategy: Manipulate and bound the rational expression.
• Steps:
  • Simplify |3/(u + 2) - 1| and work through algebraic steps.
  • Set bounds for u and find appropriate δ.
  • Show that δ satisfies the required condition.

Problem 7: Polynomial & Square Root Function

• Problem: Prove that limit as t approaches -1 of sqrt(6 - 3t) is 3.
• Strategy: Utilize algebraic simplifications and bounds.
• Steps:
  • Multiply by conjugates and simplify expressions appropriately.
  • Properly bound the terms and select δ.
  • Show the limit holds through chosen conditions.

Problem 8: Rational Function with Square Root

• Problem: Prove that limit as x approaches 5 of 7/sqrt(x - 1) is 7/2.
• Strategy: Algebraic manipulation using conjugates.
• Steps:
  • Simplify |7/sqrt(x - 1) - 7/2|, using algebraic techniques.
  • Use bounds to choose δ meeting the criteria.
  • Show compliance to the inequality for proving the limit.

Conclusion

• Understanding epsilon-delta definition is crucial for proving limits rigorously.
• Using algebraic manipulations, conjugates, and bounding techniques helps in solving diverse limit problems.
• Practice multiple problems to gain confidence in using epsilon-delta techniques.