Definition

Let ${\displaystyle A\subseteq \mathbb {R} }$ and let ${\displaystyle c\in \mathbb {R} }$

Let ${\displaystyle f,g:A\to \mathbb {R} }$

If ${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=0}$ then we say that

"As ${\displaystyle x\to c}$, ${\displaystyle f(x)=o(g(x))}$"

Examples

• As ${\displaystyle x\to \infty }$, (and ${\displaystyle m<n}$) ${\displaystyle x^{m}=o(x^{n})}$
• As ${\displaystyle x\to \infty }$, (and ${\displaystyle n\in \mathbb {N} }$) ${\displaystyle \ln x=o(x^{n})}$
• As ${\displaystyle x\to 0}$, ${\displaystyle \sin x=o(1)}$

Definition

Let ${\displaystyle A\subseteq \mathbb {R} }$ and let ${\displaystyle c\in \mathbb {R} }$

Let ${\displaystyle f,g:A\to \mathbb {R} }$

If there exists ${\displaystyle M>0}$ such that ${\displaystyle \lim _{x\to c}\left|{\frac {f(x)}{g(x)}}\right|<M}$ then we say that

"As ${\displaystyle x\to c}$, ${\displaystyle f(x)=O(g(x))}$"

Examples

• As ${\displaystyle x\to 0}$, ${\displaystyle \sin x=O(x)}$
• As ${\displaystyle x\to {\tfrac {\pi }{2}}}$, ${\displaystyle \sin x=O(1)}$

Differentiability

Let ${\displaystyle f:{\mathcal {U}}\subseteq \mathbb {R} \to \mathbb {R} }$ and ${\displaystyle x_{0}\in {\mathcal {U}}}$.

Then ${\displaystyle f}$ is differentiable at ${\displaystyle x_{0}}$ if and only if

There exists a ${\displaystyle \lambda \in \mathbb {R} }$ such that as ${\displaystyle x\to x_{0}}$, ${\displaystyle f(x)=f(x_{0})+\lambda (x-x_{0})+o\left(|x-x_{0}|\right)}$.

Mean Value Theorem

Let ${\displaystyle f:[a,x]\to \mathbb {R} }$ be differentiable on ${\displaystyle [a,b]}$. Then,

As ${\displaystyle x\to a}$, ${\displaystyle f(x)=f(a)+O(x-a)}$

Taylor's Theorem

Let ${\displaystyle f:[a,x]\to \mathbb {R} }$ be n-times differentiable on ${\displaystyle [a,b]}$. Then,

As ${\displaystyle x\to a}$, ${\displaystyle f(x)=f(a)+{\tfrac {(x-a)f'(a)}{1!}}+{\tfrac {(x-a)^{2}f''(a)}{2!}}+\ldots +{\tfrac {(x-a)^{n-1}f^{(n-1)}(a)}{(n-1)!}}+O\left((x-a)^{n}\right)}$