Limits & Continuity
The conceptual heart of the course. The ε–δ definition of a limit is the single idea everything downstream — derivatives, integrals, series — is built on. Spend real time until you can write ε–δ proofs cold.
- State the ε–δ definition of $\lim_{x\to a}f(x)=L$ precisely.
- Prove simple limits directly from the definition.
- Use the limit laws and the squeeze theorem.
- Decide whether a function is continuous at a point.
Early calculus worked remarkably well, but many infinitesimal arguments lacked a precise foundation. The ε–δ limit answered the question — what does "approaches" actually mean? — and gave analysis a rigorous framework.
Every instantaneous rate is a limit: speed on a speedometer, a tolerance an engineer designs to, whether an algorithm settles on an answer. Continuity is the hidden assumption behind root-finding by bisection and behind "small change in input, small change in output" in physics and economics.
Newton and Leibniz built calculus in the 1600s on infinitesimals; Bishop Berkeley ridiculed them in 1734 as "the ghosts of departed quantities." It took Cauchy and Weierstrass another century to make ε–δ rigorous. Weierstrass then exhibited a function that is continuous everywhere but differentiable nowhere — destroying everyone's intuition at once.
The mental model
An $\varepsilon$-$\delta$ proof is a game with two distances. The output distance $|f(x)-L|$ is what the challenger wants small. The input distance $|x-a|$ is what you control. Scratch work moves backward from the output inequality to a condition on $|x-a|$; the final proof then moves forward from your chosen $\delta$ to the desired output bound.
How to recognize the right tool
Not every limit needs a full $\varepsilon$-$\delta$ proof. The course expects you to know when a theorem is doing the work and when the definition is being tested directly.
- Direct definition. Use when asked to prove a basic limit from scratch, especially linear and polynomial examples.
- Limit laws. Use after the component limits are known and denominators do not approach $0$.
- Squeeze theorem. Use when a hard expression is trapped by simple expressions, often because $|\sin x|\le1$ or $|\cos x|\le1$.
- Continuity. Use to replace $\lim_{x\to a}f(x)$ by $f(a)$ only after checking the function is continuous at $a$.
The ε–δ definition
$\displaystyle\lim_{x\to a}f(x)=L$ means: for every $\varepsilon>0$ there exists $\delta>0$ such that, for all $x$, $$0<|x-a|<\delta\ \Longrightarrow\ |f(x)-L|<\varepsilon.$$
Read it as a challenge–response game. The challenger picks a tolerance $\varepsilon$ around $L$; you must produce a radius $\delta$ around $a$ so that every $x$ within $\delta$ (except $a$ itself) lands within $\varepsilon$ of $L$. The "$0<|x-a|$" part says the value at $a$ is irrelevant. The order of quantifiers matters: $\delta$ may depend on $\varepsilon$, but it must work for every eligible $x$.
Move ε. The red horizontal band is the required output tolerance; the blue vertical band is the input window produced by the proof's conservative choice $\delta=\min\{1,\varepsilon/5\}$.
Writing an ε–δ proof
The structure is always the same: "Let $\varepsilon>0$. Choose $\delta=\dots$. Then if $0<|x-a|<\delta$, …, so $|f(x)-L|<\varepsilon$."
Prove $\lim_{x\to3}(2x-1)=5$.
Prove $\lim_{x\to2}x^2=4$.
Limit laws and the squeeze theorem
If $\lim_{x\to a}f=L$ and $\lim_{x\to a}g=M$, then the limits of $f\pm g$, $fg$, and (if $M\ne0$) $f/g$ are $L\pm M$, $LM$, and $L/M$.
If $g(x)\le f(x)\le h(x)$ for all $x$ sufficiently close to $a$ (possibly excluding $a$), and $\lim_{x\to a}g=\lim_{x\to a}h=L$, then $\lim_{x\to a}f=L$.
$\lim_{x\to0}x^2\sin(1/x)=0$, since $-x^2\le x^2\sin(1/x)\le x^2$ and both bounds tend to $0$.
One-sided and infinite limits
$\lim_{x\to a^-}$ and $\lim_{x\to a^+}$ restrict $x$ to one side; the two-sided limit exists iff both one-sided limits exist and agree. Infinite limits and limits at infinity use the same challenge–response structure.
$\lim_{x\to a}f(x)=\infty$ means: for every $M>0$, there is a $\delta>0$ such that $0<|x-a|<\delta\Rightarrow f(x)>M$.
$\lim_{x\to\infty}f(x)=L$ means: for every $\varepsilon>0$, there is an $N>0$ such that $x>N\Rightarrow |f(x)-L|<\varepsilon$.
Continuity
$f$ is continuous at $a$ if $a$ is in the domain and $\displaystyle\lim_{x\to a}f(x)=f(a)$. Equivalently: $\forall\varepsilon>0\, \exists\delta>0$ with $|x-a|<\delta\Rightarrow|f(x)-f(a)|<\varepsilon$.
Sums, products, quotients (nonzero denominator), and compositions of continuous functions are continuous. Polynomials are continuous everywhere; rational functions wherever defined.
Let $f(x)=x\sin(1/x)$ for $x\ne0$ and $f(0)=0$. Is $f$ continuous at $0$?
If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then there is some $c\in[a,b]$ with $f(c)=N$.
$p(x)=x^3+x-1$ is continuous, $p(0)=-1$, and $p(1)=1$. By the IVT, some $c\in(0,1)$ satisfies $p(c)=0$. The theorem proves existence, not a formula for $c$.
Discontinuities commonly appear as removable holes, jump discontinuities where one-sided limits disagree, or infinite discontinuities. The ε–δ definition handles all of them without relying on a graph.
• Choosing $\delta$ that secretly depends on $x$ — it may depend only on
$\varepsilon$ (and $a$).
• Forgetting to bound the "extra" factor (like $|x+2|$) by restricting $\delta\le
1$ first.
• Confusing "limit exists" with "function is defined": the limit ignores the
value at $a$.
Practice
Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.
Let $\varepsilon>0$. Find all $\delta>0$ such that $|x-3|<\delta\Rightarrow|5x-15|<\varepsilon$.
Answer
Prove $\lim_{x\to3}(5x+1)=16$ directly from the definition.
Solution
Prove $\lim_{x\to0}(x^3+x^2)=0$ directly from the definition.
Solution
Prove: if $\lim_{x\to a}|f(x)|=0$, then $\lim_{x\to a}f(x)=0$.
Solution
Suppose $\lim_{x\to a}f(x)=0$ and $g$ is bounded near $a$. Prove $\lim_{x\to a}f(x)g(x)=0$.
Solution
Compute $\lim_{x\to\infty}\dfrac{x^2+2x+3}{3x^2+4x+5}$.
Answer
Using $\lim_{x\to0}\dfrac{\sin x}{x}=1$, compute $\lim_{x\to0}\dfrac{\sin(5x)}{x}$.
Answer
Prove that $x^4-2x=100$ has at least two solutions.
Solution
Expanded topic checklist
Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:
- Limits. Use the epsilon-delta definition, limit laws, squeeze arguments, and one-sided limits.
- Continuity. Connect continuity at a point to limits and continuity on intervals to global behavior.
- The real line. Use order, intervals, absolute value inequalities, and least-upper-bound intuition when choosing delta.
- Existence theorems. Keep IVT and EVT ready because they power root-existence and maximum-minimum arguments later.
Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.
The core chain is: limit definition gives pointwise control; continuity transfers nearby input control to output control; compact-interval theorems give existence.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Prove from the definition that $\lim_{x o2}(3x-1)=5$.
Guide
Show that $f(x)=x^2$ is continuous at $a$ using epsilon-delta reasoning.
Guide
Prove that $x^3+x-1=0$ has a root in $(0,1)$.
Guide
Give a function with $\lim_{x o0} f(x)$ not existing even though both one-sided limits are finite.