Infi-1 17

Infi-1 17

Functions #definition

\begin{matrix} We are interested in paticular functions: \\ f : A \to R, A \subseteq R \\ For example: \\ f : R ∖ {0} \to R, f (x) = \frac{7}{3 x} \end{matrix}

Limit of the function

\begin{matrix} lim_{x \to a} f (x) is an answer to the question "to where does function tend \\ when x tends to a ?" \end{matrix}

Difference from the limit of the sequence

\begin{matrix} 1. & In sequences, n \to \infty . In functions x \to a, where a \in R \cup {\infty, - \infty} \\ can be any number \\ 2. & x can tend to a from the left(smaller than a) or from the right(greater than a) \\ f (x) = {\begin{cases} 2 & x \leq 5 \\ 1 & x > 5 \end{cases} ⟹ lim_{x \to 5^{+}} f (x) = 1, lim_{x \to 5^{-}} f (x) = 2 \\ 3. & Value of f (a) does not affect the value of the limit \end{matrix}

Formal definitions of the limit #definition

Limit: Cauchy's definition

\begin{matrix} \forall ε > 0 : \exists ζ > 0 : \forall x : [| x - a | < ζ ⟹ | f (x) - L | < ε] \\ ⟹ lim_{x \to a} f (x) = L \end{matrix}

Limit: Heine's definition

\begin{matrix} L is called the limit of function f (x) \\ If, for every sequence x_{n} convergent to a, sequence f (x_{n}) converges to L \\ \forall x_{n} : [x_{n} \to a \land x_{n} \neq a ⟹ f (x_{n}) \to L] \\ ⟹ lim_{x \to a} f (x) = L \end{matrix}

\begin{matrix} f (x) = {\begin{cases} 2 & x \leq 5 \\ 1 & x > 5 \end{cases} \\ Let x_{n} = 5 - \frac{1}{n}, y_{n} = 5 + \frac{1}{n} \\ x_{n} \to 5, y_{n} \to 5 \\ \forall n : x_{n} < 5 ⟹ \forall n : f (x_{n}) = 1 \to 1 \\ \forall n : y_{n} > 5 ⟹ \forall n : f (y_{n}) = 2 \to 2 \\ ⟹ \underset{By Heine}{∄} lim_{x \to 5} f (x) \\ Let z_{n} = 5 + \frac{(- 1)^{n}}{n} \\ z_{n} \to 5 \\ f (z_{n}) = 2, 1, 2, 1, 2, 1, \dots ⟹ ∄ lim_{n \to \infty} f (z_{n}) \end{matrix}

One-sided limits of the function #definition

Right-sided limit (right limit)

\begin{matrix} L is called a right-sided limit of f \\ If, for every sequence x_{n} \to a, x_{n} > a, sequence f (x_{n}) converges to L \\ \forall x_{n} : [x_{n} \to a \land x_{n} > a ⟹ f (x_{n}) \to L] \\ ⟹ lim_{x \to a^{+}} f (x) = L \end{matrix}

Left-sided limit (left limit)

\begin{matrix} L is called a left-sided limit of f \\ If, for every sequence x_{n} \to a, x_{n} < a, sequence f (x_{n}) converges to L \\ \forall x_{n} : [x_{n} \to a \land x_{n} < a ⟹ f (x_{n}) \to L] \\ ⟹ lim_{x \to a^{-}} f (x) = L \end{matrix}

Existence of the limit of the function #lemma

\begin{matrix} \exists lim_{x \to a} f (x) = L ⟺ lim_{x \to a^{+}} f (x) = lim_{x \to a^{-}} f (x) = L \\ Proof: \\ lim_{x \to a} f (x) = L \\ ⟺ \forall x_{n} : [x_{n} \to a \land x_{n} \neq a ⟹ f (x_{n}) \to L] \\ ⟺ \forall x_{n} : [x_{n} \to a \land (x_{n} < a \lor x_{n} > a) ⟹ f (x_{n}) \to L] \\ ⟺ \forall x_{n} : [(x_{n} \to a \land x_{n} < a) \lor (x_{n} \to a \land x_{n} > a) ⟹ f (x_{n}) \to L] \\ ⟺ lim_{x \to a^{+}} f (x) = L = lim_{x \to a^{-}} f (x) \end{matrix}

Properties of the limit of the function #lemma

\begin{matrix} In general, properties of limits of the sequence also apply to the limits of the function \end{matrix}