Limits of Functions (ε–δ)
Formal definition of a limit.
Key idea
Formal definition of a limit.
In depth
lim f(x) = L as x → a means for each ε > 0 there is δ > 0 so that 0 < |x − a| < δ implies |f(x) − L| < ε. It turns "approaches" into inequalities you can prove.
Worked example
Show lim (3x + 1) = 7 as x → 2: |3x + 1 − 7| = 3|x − 2| < ε if δ = ε/3.
Formula
Real-world uses
- Proving rigorously that a function is accurate enough near a point, as in engineering tolerances
Common mistake
Letting δ depend on x instead of only on ε and a.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Limits of Functions (ε–δ)?
Formal definition of a limit. It is a topic at the expert level in Analysis & Topology.
What is the formula for Limits of Functions (ε–δ)?
The key formula is |f(x) − L| < ε when 0<|x−a|<δ.
Where is Limits of Functions (ε–δ) used in real life?
Real-world examples: Proving rigorously that a function is accurate enough near a point, as in engineering tolerances.
What should I learn before Limits of Functions (ε–δ)?
Make sure you are comfortable with Rearrangement of Series, the previous topic in Analysis & Topology.
What is a common mistake with Limits of Functions (ε–δ)?
Letting δ depend on x instead of only on ε and a.
How can I practise Limits of Functions (ε–δ)?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Limits of Functions (ε–δ)?
Move on to Continuity (ε–δ) next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.