Infi-1 19

\begin{matrix} f (x), g (x) \\ f (x) \to 1 ⟹ lim f (x)^{g (x)} = e^{lim g (x) (f (x) - 1)} \end{matrix}

\begin{matrix} lim_{x \to 0} (\cos x)^{1 / x} = e^{lim_{x \to 0} (\cos x - 1) / x} = e^{lim_{x \to 0} - (1 - \cos x) / x} = e^{0} = 1 \\ ? ? ? \end{matrix}

\begin{matrix} If f is continuous at point a, then lim_{x \to a} f (x) = f (a) \end{matrix}

\begin{matrix} lim_{x \to 0} \frac{x^{2} + 7 \cdot 9^{x^{3}}}{\tan x + \cot x} = \frac{0 + 7 \cdot 9^{0^{3}}}{\tan 0 + \cot 0} = 7 \end{matrix}

\begin{matrix} By definition of limit of function: \\ lim_{x \to a} f (x_{n}) = f (a) \\ \forall x_{n} : x_{n} \to a, x_{n} \neq a : f (x_{n}) \to f (a) \end{matrix}

Continuity of functions #definition

\begin{matrix} lim_{x \to a} f (x) = L ⟹ g (lim_{x \to a} f (x)) = g (L) \end{matrix}

\begin{matrix} g continuous at a and f continuous at g (a) \\ ⟹ (f \circ g) is continuous at a \end{matrix}

\begin{matrix} f is called continuous on [a, b] iff \\ 1. \forall c \in (a, b) : f is continuous at c \\ 2. lim_{n \to a^{+}} f (x) = f (a) \\ 3. lim_{x \to b^{-}} f (x) = f (b) \end{matrix}

\begin{matrix} a is called a discontinuity of f iff f is not continuous at a \end{matrix}

\begin{matrix} \exists lim_{x \to a} f (x) \\ lim_{x \to a} f (x) \neq f (a) \end{matrix}

\begin{matrix} \exists lim_{x \to a^{-}} f (x), \exists lim_{x \to a^{+}} f (x) \\ lim_{x \to a^{-}} f (x) \neq lim_{x \to a^{+}} f (x) \end{matrix}

\begin{matrix} ∄ lim_{x \to a^{-}} f (x) \lor ∄ lim_{x \to a^{+}} f (x) \end{matrix}