Limits and continuity

First year undergraduate foundational mathematics, calculus, limits and continuity of functions.

What is the formal definition of a limit?

$$\lim_{x\rightarrow a}f\left(x\right)=L$$

if for every number \varepsilon>0 there is a number \delta>0 such that

if 0<\left|x-a\right|<\delta then\left|f\left(x\right)-L\right|<\varepsilon

What is meant by the limit of a function at a point x?

The value the function approaches as x gets closer and close to a specific value.

$$\lim_{x\rightarrow c}f\left(x\right)=L$$

Finding a limit rules

• $$\lim_{x\to c}x=c$$

• L’Hopital’s $\lim_{x\to c}\frac{f(x)}{g(x)}=\frac{f^{\prime}(x)}{g^{\prime}(x)}$ (for indeterminants)

Definition of the left-hand limit

$$\lim_{x\to a^{-}}f\left(x\right)=L$$

if for every number \varepsilon>0 there is a number \delta>0 such that

if a-\delta<x<a then \left|f\left(x\right)-L\right|<\varepsilon

Definition of the right-hand limit

$$\lim_{x\to a^{+}}f\left(x\right)=L$$

if for every number \varepsilon>0 there is a number \delta>0 such that

if a<x<a+\delta then \left|f\left(x\right)-L\right|<\varepsilon

What is the squeeze/sandwich theorem

if $f\left(x\right)\le g\left(x\right)\le h\left(x\right)$ and

$\lim_{x\rightarrow c}f\left(x\right)=L$ and $\lim_{x\rightarrow c}fh\left(x\right)=L$ then

$$\lim_{x\rightarrow c}g\left(x\right)=L$$

When is a function said to be continuous?

A function is continuous at a point c when

$f\left(x\right)\rightarrow f\left(c\right)$ as $x\rightarrow c$ or $\lim_{x\to c}f\left(x\right)=f\left(c\right)$i.e. when:

• f is defined at c

• The limit of f(x) at x=c exists

• $\lim_{x\to c}f\left(x\right)=f\left(c\right)$
  

The function is continuous everywhere if it is continuous at every point c on its domain.

When is a function continuous from the right?

if $\lim_{x\to a+}f\left(x\right)=f\left(a\right)$

When is a function continuous from the left?

$$\lim_{x\to a-}f\left(x\right)=f\left(a\right)$$

When is a function continuous on an interval?

If it is continuous at every number in the interval

Polynomial continuity rule

A polynomial is continuous everywhere on

$$\mathbb{R}=\left(-\infty,\infty\right)$$

What functions are continuous everywhere on their domains?

• polynomials

• rational functions

• root functions

• trigonometric

• inverse trigonometric

• exponential

• logarithmic

Continuity of combination/composition functions

• For composition: f must be continuous at c and g must be continuous at $f\left(c\right)$ for $g\circ f$ to be continuous at c

• For combination: f and g must both be continuous at c

What is the intermediate value theorem for continuity

Let f be continuous on the closed interval [a,b] and let $f\left(a\right)\le N\le f\left(b\right)$ where $f\left(a\right)\ne f\left(b\right)$.

Then there exists a number c in (a,b) such that $f\left(c\right)=N$

What is the intermediate value theorem (limits)

If d is any real number such that

$f\left(a\right)\le d\le f\left(b\right)$ or $f\left(b\right)\le d\le f\left(a\right)$ then there exists $c\in\left\lbrack a,b\right\rbrack$ such that $f\left(c\right)=d$

What is Bolzano’s theorem

If f(a) and f(b) have opposite signs then there exists a $c\in\left(a,b\right)$ such that $f\left(c\right)=0$

Precise definition of infinite limits at a point

$$\lim_{x\rightarrow a}f\left(x\right)=\infty$$

for every positive number M there is a positive number $\delta$ such that

if 0<\left|x-a\right|<\delta then

f\left(x\right)>M.

Precise definition for infinite limits at a point (negative infinity)

$$\lim_{x\rightarrow a}f\left(x\right)=-\infty$$

if for every negative number N there is a positive number $\delta$ such that

if 0<\left|x-a\right|<\delta then f\left(x\right)<N