Functions, Limits, and Continuity
What it means for a function to have a limit, why continuity is the right notion of ‘no sudden jumps’, and how the epsilon-delta idea makes both precise.
Leads to: Derivatives (2.2) are limits of difference quotients; continuity is the hypothesis behind the Intermediate and Extreme Value Theorems.
Learning Objectives
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- State the epsilon-delta definition of a limit and interpret each quantifier geometrically.
- Compute limits using algebraic simplification and the limit laws.
- Define continuity at a point and on an interval and classify basic discontinuities.
- Explain the Intermediate Value Theorem and use it to locate a root.
Key Vocabulary
- Limit
- L is the limit of f at c if f(x) can be forced arbitrarily close to L by taking x close enough (but not equal) to c.
- Epsilon-delta
- ∀ε>0 ∃δ>0: 0<|x−c|<δ ⇒ |f(x)−L|<ε.
- Continuity at c
- f is continuous at c when lim_{x→c} f(x)=f(c): the limit exists and equals the value.
- Removable / jump discontinuity
- A hole that a single redefinition would fix / a genuine step where left and right limits differ.
- One-sided limit
- The limit taken as x approaches c only from the left (c−) or right (c+).
- Intermediate Value Theorem
- A continuous f on [a,b] attains every value between f(a) and f(b).
From sequences to functions
The limit of a function generalizes the sequence limit. We say \(\lim_{x\to c}f(x)=L\) if we can make \(f(x)\) within any tolerance \(\varepsilon\) of \(L\) by keeping \(x\) within some \(\delta\) of \(c\) (excluding \(c\) itself):
The exclusion \(0\lt |x-c|\) matters: the limit is about the approach to \(c\), not the value at \(c\). A function can have a limit at a point where it is not even defined.
Computing limits
Most limits are found by the limit laws (limits of sums, products, quotients split apart when the pieces exist) plus algebra to remove a \(0/0\) form. For example \(\frac{x^2-1}{x-1}=x+1\) for \(x\ne1\), so \(\lim_{x\to1}\frac{x^2-1}{x-1}=2\).
Continuity and its guarantees
A function is continuous at \(c\) when the limit exists and equals the value, \(\lim_{x\to c}f=f(c)\). Continuity on an interval unlocks two cornerstone theorems: the Intermediate Value Theorem (a continuous function skips no values) and the Extreme Value Theorem (a continuous function on a closed bounded interval attains a max and a min - the basis of optimization in 2.5).
Interactive: see the limit / spot the discontinuity
- Assuming lim_{x→c} f = f(c) always; that equality IS continuity and can fail (removable or jump discontinuity).
- Plugging in to a 0/0 form and declaring ‘no limit’; 0/0 is indeterminate - simplify first.
- Forgetting the strict 0 < |x − c| - the limit ignores the value at c entirely.
- Applying the quotient limit law when the denominator’s limit is 0.
- For a 0/0 rational limit, factor and cancel the common root; for roots, multiply by the conjugate.
- Check one-sided limits separately when a piecewise definition or absolute value is involved.
- Use the IVT to guarantee a root before you hunt for it numerically - sign change on [a,b] suffices.
- Continuity of elementary functions (polynomials, exp, sin, cos) is a fact you may cite; only the joins need checking.
Practice this in the Euler Lab
Computational problems that exercise exactly this technique. Each opens in the Euler Lab with a Python workbench, a progressive hint ladder, and answer checking. Tier A/B run at full scale in the browser.
Warm-up:
#16 Power Digit Sum (1%, tier A) #30 Digit Fifth Powers (2%, tier A) #52 Permuted Multiples (2%, tier A) #33 Digit Cancelling Fractions (3%, tier A)
Applied:
#932 $2025$ (17%, tier B) #315 Digital Root Clocks (18%, tier B) #479 Roots on the Rise (19%, tier B)
Challenge:
#666 Polymorphic Bacteria (41%, tier C) #157 Base-10 Diophantine Reciprocal (42%, tier C)
130 Project Euler problems in total are mapped to this lesson. Open the Euler Lab to filter them all.
Knowledge Check
Practical Exercise
Show that \(f(x)=\dfrac{\sqrt{x+1}-1}{x}\) has a limit as \(x\to0\) and find it. Is \(f\) continuous at 0? Explain.
Direct substitution gives \(0/0\). Multiply by the conjugate:
As \(x\to0\) this tends to \(\tfrac{1}{1+1}=\tfrac12\). So the limit is \(1/2\). However \(f(0)\) is undefined (division by zero), so \(f\) is not continuous at 0 - it has a removable discontinuity, patchable by defining \(f(0)=1/2\).
Lesson Summary
Formula Sheet Additions
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: The limit describes the approach to c (0<|x−c|<δ), never the value at c; so a removable hole leaves the limiting behavior intact, as in (x²−4)/(x−2) at 2.
A: Continuity at c means lim_{x→c} f(x)=f(c); it enables the Intermediate Value Theorem (continuous functions attain all intermediate values) and the Extreme Value Theorem.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Calculus, Vol. I (Tom Apostol, 2nd ed., 1967) foundational - Vol. I, Ch. 3 - Rigorous limits, continuity, and the Intermediate/Extreme Value Theorems.
- A Mind for Numbers (Barbara Oakley, 2014) foundational - learning hard concepts - Chunking the epsilon-delta idea and overcoming the intimidation of the first rigorous definition.