Limit (Function)

analysis

Definition

Limit (Function)

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a function. The limit of $f$ as $x$ approaches $x_0$ is equal to $L$ if, for every $\epsilon >0$, there exists a $\delta >0$ such that for all $x\in D$:

$$0<|x-x_0|<\delta ⟹|f(x)-L|<\epsilon$$

This is denoted as ${\lim }_{x\to x_0}f(x)=L$.

Sequential Definition

Alternatively, the limit can be defined using sequences (Heine’s definition):

$$\underset{x\to x_0}{\lim }f(x)=L⟺\forall (x_n{)}_{n\in \mathbb{N}}\subseteq D\setminus \{x_0\}:\left(\underset{n\to \infty }{\lim }{x}_n=x_0⟹\underset{n\to \infty }{\lim }f(x_n)=L\right)$$

Properties

• Uniqueness: If a limit exists, it is unique.
• Algebra of Limits: Limits distribute over addition, subtraction, multiplication, and division (provided the denominator is non-zero).
• Squeeze Theorem: If $g(x)\le f(x)\le h(x)$ and $\lim g(x)=\lim h(x)=L$, then $\lim f(x)=L$.