Limits

Limits of functions:

There are many ways to interpret a function:

• As a formula

$f(x)=x^2-1$

• As a machine model

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• As an arrow diagram

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• As a graph

Limits

Let $f:D\to \mathbb{R}$ be a function with domain $D\subseteq \mathbb{R}$.

We say $\underset{x\to c}{lim}f(x)=L$

if f(x) tends to L whenever x tends to c.

• the limit point c does not need to be an element of D
• while x approaches c (but is never equal to c), it must be an element of D.

Replacement info

If $f(x)=g(x)\text{ for all }x\ne c\text{ then }\underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}g(x)$

One-sided limits

We say

$li{m}_{x\to c^{-}}f(x)=L$

if f(x) tends to L when x approaches c from the left.

$li{m}_{x\to c^{+}}f(x)=L$

if f(x) tends to L whenever x approaches c from the right.

info

• $\text{If }\underset{x\to c}{lim}f(x)=L\text{ exists, then }\underset{x\to c^{+}}{lim}f(x)=\underset{x\to c^{-}}{lim}f(x)=L\text{ exist}$
• $\text{If }\underset{x\to c^{-}}{lim}f(x)\text{ and }\underset{x\to c^{-}}{lim}f(x)=L\text{ exist, then }\underset{x\to c}{lim}f(x)=L\text{ exists}$

• The limit lim does not exist if one of the following is true:
  • the left limit does not exist. -
  • the right limit does not exist. +
  • both the left and right limit exist but they are not equal.

Limits to infinity

Definition

We say $\underset{x\to \mathrm{\infty }}{lim}f(x)=L$ if f(x) tends to L whenever x tends to $\mathrm{\infty }$, and

$\underset{x\to -\mathrm{\infty }}{lim}f(x)=L$ if f(x) tends to L whenever x tends to $-\mathrm{\infty }$

WARNING

Even if $\underset{x\to c}{lim}=+-\mathrm{\infty }$, then the limit does not exist!

Well behaved functions

well behaved functions:

• polynomials
• exponential functions
• logarithms
• sine
• cosine

because a limit to a point c in the domain of f can be calculated by direct substitution.

$\underset{x\to c}{lim}f(x)=f(c)$

Replacement Rule

info (weak)

If $f(x)=g(x)forallx\ne c,then\underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}g(x)$

info (strong)

Let I be an open interval containing c. If $f(x)=g(x)forallx\in I$, then $\underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}g(x)$

Limit Laws 📜

Assume that both $\underset{x\to c}{lim}f(x)=L$ and $\underset{x\to c}{lim}g(x)=M$ exist.

Name	Rule
Sum Rule	$\underset{x\to c}{lim}(f(x)+g(x))=L+M$
Difference Rule	$\underset{x\to c}{lim}(f(x)-g(x))=L-M$
Constant Multiple Rule	$\underset{x\to c}{lim}(k\cdot f(x))=k\cdot L$
Product Rule	$\underset{x\to c}{lim}(f(x)\cdot g(x))=L\cdot M$
Quotient Rule	$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\frac{L}{M},M\ne 0$
Power Rule	$\underset{x\to c}{lim}(f(x){)}^n=L^n,\text{ with n }\in \mathbb{N}$
Root Rule	$\underset{x\to c}{lim}\sqrt{f(x)}=\sqrt{L}=L^{\frac{1}{n}}\text{, with n }\in \mathbb{N}$

Indeterminate forms:

A limit is called an indeterminate form if applying the limit laws leads to an indecisive result.

limit law	Indeterminate Situation
Difference Rule	$\mathrm{\infty }-\mathrm{\infty }$
Product Rule	$0\cdot infty$
Quotient Rule	$\frac{\mathrm{\infty }}{\mathrm{\infty }}or\frac{0}{0}$
Power Rule	$1^{\mathrm{\infty }},0^{0}or{\mathrm{\infty }}^0$

The Conjugate Trick

• The conjugate trick is based on the following identity: (a + b)(a - b) = a^2 - b^2
• a + b is the conjugate of a - b (and vice versa)

The sandwich info

• Let f, g and h functions such that $g(x)\le f(x)\le h(x)\text{ for all }x\ne c$
• f is "sandwiched" between g and h.

info

If $\underset{x\to c}{lim}g(x)and\underset{x\to c}{lim}h(x)$ exist, and moreover, are equal (say to L), then $\underset{x\to c}{lim}f(x)$ exists, and is equal to L

Continuity at a point.

Definition

Let $f:[a,b]\to \mathbb{R}$ be a function. Let $c\in [a,b]$. Interior Points:

if a < c < b, then f is continuous at c if $\underset{x\to c}{lim}f(x)=f(c)$

Endpoints:

if c = a or c = b, then f is continuous at a if $\underset{x\to c}{lim}f(x)=f(a)$

and f is continuous at b if $\underset{x\to c}{lim}f(x)=f(b)$

Practical approach 🔍

Continuity Test

A function f(x) is continuous at an interior point c of its domain if and only if

1. f(c) exists -> c lies in the domain of f.
2. $\underset{x\to c}{lim}f(x)$ exists -> f has a limit as x approaches c.
3. $\underset{x\to c}{lim}f(x)=f(c)$ -> the limit equals the function value.

• If one (or more) of the conditions is not satisfied, f is not continuous at c.

Discontinuities:

In all following cases f is not continuous at c:

Functions	c	Violation
$f(x)=\frac{x^2-1}{x-1}$	1	f is not defined at c

Laws of continuity 📜

Assume that f and g are continuous at c, then the following combinations are continuous at c.

Name	Rule
Sums	$f+g$
Differences	$f-g$
Constant Multiples	$k\cdot f,withk\in \mathbb{R}$
Products	$f\cdot g$
Quotients	$\frac{f}{g},g(c)\ne 0$
Powers	$f^n,withn\in \mathbb{N}$
Roots	$\sqrt{f},withn\in \mathbb{N}$

Composition of continuous functions:

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info

If f is continuous at c, and g is continouous at f(c), then $g\circ f$ is continous at c.

• the composition $g\circ f$ is the function that maps x to g(f(x))

Global continuity

Definition

• Let I be an interval in $\mathbb{R}$. A function f is continuous on I if for all $c\in I$ the function f is continuous in c.
• A function f is continuous if f is continuous on its domain.

Formula functions

Definition

A formula function is a function constructed from elementary functions:

• polynomials
• power functions
• trig functions
• exp functions
• logarithms

and using algebraic operations like:

• add
• subtraction
• multiplication
• division
• composition

• All formula functions are continuous.
• For all formula functions f and for all $c\in Dom(f)$: $\underset{x\to c}{lim}f(x)=f(c)$