Limit

Limit

$\underset{x\to a}{\lim }f(x)=b$ if $\forall ϵ>0,\exists \delta >0$ such that $|x-a|<\delta$ implies $|f(x)-b|<ϵ$.

Upper Bound and Lower Bound

For a set \require{mhchem}\newcommand{\R}{\mathbb{R}}\newcommand{\N}{\mathbb{N}}S\subseteq \R, if exists $b\in \mathbb{R}$ such that for all $s\in S$, $s\le b$, then $b$ is an upper bound for $S$. Similarly, if for all $s\in S$, $s\ge b$, then $b$ is called a lower bound.

Supremum and Infimum

$b$ is called supremum of a set $S$ if $b$ is an upper bound and for all upper bound $u$, $b\le u$. Similarly, $b$ is infimum of $S$ if it is the greatest upper bound.

Lemma

Lemma

• $b=\sup (S)$ iff for all $\epsilon >0$ there exists $x\in S$ such that $b-x<\epsilon$.
• $b=\inf (S)$ iff for all $\epsilon >0$ there exists $x\in S$ such that $x-b<\epsilon$.

Maximum and Minimum

Suppose $S\subseteq \mathbb{R}$. If there exists $x\in S$ such that $x$ is supremum of $S$, then $x$ is called the maximum of $S$. Similarly, $x$ is called the minimum if $x$ is the infimum lying in $S$.

L'Hôpital's Rule

Let $f$ and $g$ be functions differentiable on an open interval $I$ containing $a$ (except possibly at $a$ itself), and suppose:

• $\underset{x\to a}{\lim }f(x)=\underset{x\to a}{\lim }g(x)=0$ or $\pm \infty$,
• The limit $\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists.

Then,

$$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}.$$