Theory

Limits and continuity

What it means for a function to tend to a value, how limits are computed, which theorems govern them, and what continuity adds.

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Neighborhoods and accumulation points
Definition of limit

Neighborhoods and accumulation points

1. Neighborhood of a point

For $x_{0} \in R$ and $r > 0$ , the neighborhood of $x_{0}$ of radius $r$ is the open interval

I (x_{0}, r) = (x_{0} - r, x_{0} + r) = {x \in R : ∣ x - x_{0} ∣ < r} .

A neighborhood of $+ \infty$ is a half-line $(a, + \infty)$ ; a neighborhood of $- \infty$ is a half-line $(- \infty, a)$ . The expressions $∣ x - x_{0} ∣ < r$ and $x \in (x_{0} - r, x_{0} + r)$ say the same thing, and both appear in the definitions of limit.

2. Accumulation point

$x_{0} \in R$ is an accumulation point of $A \subseteq R$ if every neighborhood of $x_{0}$ contains at least one point of $A$ other than $x_{0}$ . The point may or may not belong to $A$ . The limit as $x \to x_{0}$ makes sense only at an accumulation point of the domain.

$A = (0, 1)$ : the accumulation points are all the points of $[0, 1]$ , including $0$ and $1$ , which do not belong to $A$ .
$A = {1, 2, 3}$ : no accumulation points.
$A = N$ : no accumulation points in $R$ ; only $x \to + \infty$ makes sense.

Definition of limit

1. Finite limit at a point

$f : A \to R$ , $x_{0}$ an accumulation point of $A$ . We write $x \to x_{0} lim f (x) = L$ , with $L \in R$ , if for every $ε > 0$ there exists $δ > 0$ such that, for every $x \in A$ ,

0 < ∣ x - x_{0} ∣ < δ ⟹ ∣ f (x) - L ∣ < ε .

Given a distance $ε$ from $L$ , as small as desired, there exists a neighborhood of $x_{0}$ in which all the values $f (x)$ lie at a distance less than $ε$ from $L$ ; the radius $δ$ depends on $ε$ . The condition $0 < ∣ x - x_{0} ∣$ excludes $x_{0}$ , so the value $f (x_{0})$ , even when it exists, plays no role in the limit.

Verification using the definition

$x \to 2 lim (3 x - 1) = 5$ . Given $ε > 0$ , since $∣ (3 x - 1) - 5∣ = 3∣ x - 2∣$ , it suffices to take $δ = \frac{ε}{3}$ : if $∣ x - 2∣ < \frac{ε}{3}$ then $3∣ x - 2∣ < ε$ .

y = 3 x - 1

With

ε = \frac{3}{2}

and

δ = \frac{ε}{3} = \frac{1}{2}

: for

∣ x - 2∣ < δ

the graph of

y = 3 x - 1

stays within the strip

∣ y - 5∣ < ε

(shaded box)

The limit does not depend on the value at the point

$f (x) = \frac{sin x}{x}$ is defined for $x \neq = 0$ and $0$ is an accumulation point of the domain. As $x \to 0$ we have $f (x) \to 1$ (fundamental sine limit), even though $f (0)$ does not exist: the graph has a hole at $(0, 1)$ .

y = \dfrac{\sin x}{x}

y = \frac{sin x}{x}

: the point

(0, 1)

does not belong to the graph

2. Infinite limit and limit at infinity

Same structure, with neighborhoods of a different kind.

$x \to x_{0} lim f (x) = + \infty$ if for every $M > 0$ there exists $δ > 0$ such that $0 < ∣ x - x_{0} ∣ < δ ⟹ f (x) > M$ .
$x \to + \infty lim f (x) = L$ (with $A$ unbounded above) if for every $ε > 0$ there exists $K > 0$ such that $x > K ⟹ ∣ f (x) - L ∣ < ε$ .
$x \to + \infty lim f (x) = + \infty$ if for every $M > 0$ there exists $K > 0$ such that $x > K ⟹ f (x) > M$ .

For $- \infty$ one writes $f (x) < - M$ or $x < - K$ . In all cases: for every neighborhood $V$ of the limit value there exists a neighborhood $U$ of $x_{0}$ such that $f (x) \in V$ for every $x \in U \cap A$ , $x \neq = x_{0}$ .

Example

$f (x) = \frac{1}{x ^{2}}$ . As $x \to 0$ : given $M > 0$ , $\frac{1}{x ^{2}} > M$ for $0 < ∣ x ∣ < \frac{1}{M}$ , therefore $f (x) \to + \infty$ . As $x \to + \infty$ : given $ε > 0$ , $\frac{1}{x ^{2}} < ε$ for $x > \frac{1}{ε}$ , therefore $f (x) \to 0$ .

y = \dfrac{1}{x^{2}}

y = \frac{1}{x ^{2}}

: the values grow without bound as

x \to 0

and approach

0

x \to \pm \infty

3. Right-hand limit and left-hand limit

The right-hand limit $x \to x_{0}^{+} lim f (x)$ is the limit of the restriction of $f$ to the points $x > x_{0}$ ; the left-hand limit $x \to x_{0}^{-} lim f (x)$ is the limit of its restriction to the points $x < x_{0}$ . If $x_{0}$ is an accumulation point from both sides, the limit exists if and only if the two one-sided limits exist and coincide. They are needed when the function has different expressions on the right and on the left, or when a denominator changes sign at the point.

Example

$x \to 0^{+} lim \frac{1}{x} = + \infty$ and $x \to 0^{-} lim \frac{1}{x} = - \infty$ , therefore the limit as $x \to 0$ does not exist.

y = \dfrac{1}{x}

y = \frac{1}{x}

: the two one-sided limits at

0

are different

The theory continues with 7 more sections (first theorems on limits, operations with limits and indeterminate forms, fundamental limits…), plus all worked examples, the solutions and the PDF.

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