Theory

Limits and continuity

What it means for a function to approach a value, how limits are computed and indeterminate forms resolved, how asymptotes are found, what a continuous function is and what the intermediate zero theorem and the Weierstrass theorem guarantee.

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Finite limit at a point
Infinite limits and limits at infinity

Finite limit at a point

Let $f$ be defined on an open interval containing $x_{0}$ , except possibly at the point $x_{0}$ itself. We write

x \to x_{0} lim f (x) = L

if for every $ε > 0$ there exists $δ > 0$ such that, for every $x$ in the domain,

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

In other words, $f (x)$ is as close to $L$ as we like, within the tolerance $ε$ , provided $x$ is close enough to $x_{0}$ , within the distance $δ$ . The condition $0 < ∣ x - x_{0} ∣$ excludes the point $x_{0}$ , so the value $f (x_{0})$ , if it exists, does not matter. For $x \to 1 lim (2 x + 3) = 5$ , once $ε$ is fixed, we have $∣ (2 x + 3) - 5∣ = 2 ∣ x - 1∣$ , which is less than $ε$ when $∣ x - 1∣ < \frac{ε}{2}$ ; therefore $δ = \frac{ε}{2}$ suffices.

y = 2x + 3

With

ε = 1

and

δ = \frac{1}{2}

, for

0 < ∣ x - 1∣ < \frac{1}{2}

the value

2 x + 3

lies in the strip

4 < y < 6

Right-hand limit and left-hand limit

If in the definition of limit only the $x > x_{0}$ are considered, we obtain the right-hand limit, $x \to x_{0}^{+} lim f (x)$ ; with only the $x < x_{0}$ we obtain the left-hand limit, $x \to x_{0}^{-} lim f (x)$ . The limit exists if and only if the right-hand limit and the left-hand limit exist and are equal. They are needed for piecewise-defined functions, at the points where one formula switches to the other. For the function equal to $5$ for $0 < x \leq 2$ and to $8$ for $x > 2$ , the left-hand limit at $2$ is $5$ and the right-hand limit is $8$ , so the limit at $2$ does not exist.

The function equals

5

for

0 < x \leq 2

and

8

for

x > 2

, so at

x = 2

the left-hand limit is

5

and the right-hand limit is

8

Infinite limits and limits at infinity

The other situations are described in the same way, replacing the neighborhoods of a number with half-lines.

$x \to x_{0} lim f (x) = + \infty$ if for every $M$ there exists $δ > 0$ such that $f (x) > M$ for $0 < ∣ x - x_{0} ∣ < δ$ , that is, if near $x_{0}$ the values of $f$ exceed any threshold. The limit $- \infty$ is defined in the same way.
$x \to + \infty lim f (x) = L$ if for every $ε > 0$ there exists $N$ such that $∣ f (x) - L ∣ < ε$ for $x > N$ , that is, if for large values of the variable the values of $f$ stay close to $L$ . The limit as $x \to - \infty$ is defined in the same way.
$x \to + \infty lim f (x) = + \infty$ if for every $M$ there exists $N$ such that $f (x) > M$ for $x > N$ .

For example $\frac{1}{x} \to 0$ as $x \to + \infty$ , $\frac{1}{x ^{2}} \to + \infty$ as $x \to 0$ , while $\frac{1}{x}$ tends to $+ \infty$ as $x \to 0^{+}$ and to $- \infty$ as $x \to 0^{-}$ . In economic models the limit as $x \to + \infty$ describes what happens for large quantities; for example the average cost $\frac{200}{q} + 12$ tends to $12$ as $q \to + \infty$ .

y = \dfrac{1}{x}

\frac{1}{x}

tends to

0

x \to \pm \infty

, so the

x

-axis is a horizontal asymptote; it tends to

+ \infty

x \to 0^{+}

and to

- \infty

x \to 0^{-}

The theory continues with 5 more sections (rules of computation and indeterminate forms, standard limits and comparison of infinities, asymptotes…), plus all worked examples, the solutions and the PDF.

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