Limit of a Function

Definition

Function of a Single Variable

Epsilon-Delta-Definition of Limit

Suppose $f:\mathcal{D}\to \mathbb{R},a\in \mathcal{D},L\in \mathbb{R}$ $\forall ϵ>0,\exists \delta >0,\forall x\in \mathcal{D},(0<|x-a|<\delta \Rightarrow |f(x)-L|<ϵ)\iff \underset{x\to a}{\lim }f(x)=L$

The limit of $f$, as $x$ approaches $a$, is $L$, or say the $f$ converges to $L$, otherwise, $f$ diverges

Right-Sided Limit

Suppose $f:\mathcal{D}\to \mathbb{R},a\in \mathcal{D},L\in \mathbb{R}$ $\forall ϵ>0,\exists \delta >0,\forall x\in \mathcal{D},(0<x-a<\delta \Rightarrow |f(x)-L|<ϵ)\iff \underset{x\to a^{+}}{\lim }f(x)=f(a^{+})=L$

The right-sided limit of $f$, as $x$ approaches $a$ from the right side, is $L$

Left-Sided Limit

Suppose $f:\mathcal{D}\to \mathbb{R},a\in \mathcal{D},L\in \mathbb{R}$ $\forall ϵ>0,\exists \delta >0,\forall x\in \mathcal{D},(0<a-x<\delta \Rightarrow |f(x)-L|<ϵ)\iff \underset{x\to a^{-}}{\lim }f(x)=f(a^{-})=L$

The left-sided limit of $f$, as $x$ approaches $a$ from the left side, is $L$

Limits at infinity

Suppose $f:\mathbb{R}\to \mathbb{R}$ and $a,L\in \mathbb{R}$ $\forall ϵ>0,\exists N\in \mathbb{R},\forall x\in \mathbb{R},(x\ge N\Rightarrow |f(x)-L|<ϵ)\iff \underset{x\to \infty }{\lim }f(x)=L$

Infinite Limits

Suppose $f:\mathbb{R}\to \mathbb{R}$ and $a,L\in \mathbb{R}$ $\forall M>0,\exists \delta >0,\forall x\in \mathbb{R},(0<|x-a|<\delta \Rightarrow f(x)>M)\iff \underset{x\to a}{\lim }f(x)=\infty$

Suppose $f:\mathbb{R}\to \mathbb{R}$ and $a,L\in \mathbb{R}$ $\forall M>0,\exists N\in \mathbb{R},\forall x\in \mathbb{R},(x\ge N\Rightarrow f(x)>M)\iff \underset{x\to \infty }{\lim }f(x)=\infty$

Function of Multi Variables

Ordinary Limits

Suppose $f:\mathcal{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^m,\mathbf{a}\in \mathcal{D},\mathbf{L}\in {\mathbb{R}}^m$ $\forall ϵ>0,\exists \delta >0,\forall \mathbf{x}\in \mathcal{D},(0<||\mathbf{x}-\mathbf{a}||<\delta \Rightarrow ||f(\mathbf{x})-\mathbf{L}||<ϵ)\iff \underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})=\mathbf{L}$ where $||\mathbf{x}-\mathbf{a}||$ denotes the Euclidean distance between $\mathbf{x}$ and $\mathbf{a}$ in ${\mathbb{R}}^n$.

Properties

Algebraic Limit Theorem

Suppose $f,g:\mathcal{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$, and $\mathbf{a}\in \mathcal{D}$ Then, the following are satisfied

• $\underset{\mathbf{x}\to a}{\lim }f(\mathbf{x})+g(\mathbf{x})=\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})+\underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})$
• $\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})-g(\mathbf{x})=\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})-\underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})$
• $\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})\cdot g(\mathbf{x})=\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})\cdot \underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})$
• $\underset{\mathbf{x}\to \mathbf{a}}{\lim }\frac{f(\mathbf{x})}{g(\mathbf{x})}=\frac{\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})}{\underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})}$ where $\underset{\mathbf{x}\to \mathbf{a}}{\lim }g(\mathbf{x})\ne 0$
• $\underset{\mathbf{x}\to \mathbf{a}}{\lim }[f(\mathbf{x}){]}^n=[\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x}){]}^n$ where $n\in \mathbb{N}$
• $\underset{x\to \mathbf{a}}{\lim }[f(\mathbf{x}){]}^{1/n}=[\underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x}){]}^{1/n}$ where $n\in \mathbb{N}$

Facts

Suppose $f:\mathcal{D}\to \mathbb{R}$, and $a\in \mathcal{D}$ If $\underset{x\to a}{\lim }f(x)$ converges, then the limit is unique

Squeeze Theorem for Functions

Suppose $f,g,h:\mathcal{D}\to \mathbb{R}$, $L\in \mathbb{R}$, and $\forall x\in \mathcal{D}-\{a\},f(x)\le g(x)\le h(x)$

If $\underset{x\to a}{\lim }f(x)=\underset{x\to a}{\lim }h(x)=L$, then $\underset{x\to a}{\lim }g(x)=L$

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