Limits

Definitions

If $x$ approaches closely to some value $a$ but not equals to it, we denote this as $x \to a$ . When it approaches only by values that are more than $a$ we say that it does it from the right $x \to a^+$ . For less than $a$ values it approaches from the left $x \to a^-$ .

If $f(x) \to R$ when $x \to a^+$ there is a right-hand limit. $\lim_{x \to a^+} f(x) = R$ If $f(x) \to L$ when $x \to a^-$ there is a left-hand limit. $\lim_{x \to a^-} f(x) = L$ If $R = L$ then $f(x)$ has a limit. $\lim_{x \to a} f(x) = L = R$

Formal definition

If $\forall \epsilon>0 \; \exists \delta>0$ such that if $0<|x - \delta|<a$ then $|f(x) - L|<\epsilon$

no-limit No limit. Left-hand != Right-hand Limit exists but function is discontinuous in point

Limit Laws $\lim (f\pm g) = \lim f \pm \lim g$ $\lim (f*g) = \lim f * \lim g$ $\lim (f/g) = \lim f / \lim g, \lim g \neq 0$ $\lim (f/g) = DNE, \lim f \neq 0, \lim g = 0$ $\lim (f/g) = ?, \lim f = 0, \lim g = 0$

Function is continuous in the point $a$ when $\lim_{x \to a} f(x) = f(a)$ . Function is continuous if it’s continuous for any value of its argument. if $f$ is continuous in $b$ $\lim_{x \to a} f(g(x)) = f(b), \lim_{x \to a} g(x) = b$ Intermediate Value Theorem. If $f$ is a function which is continuous on the interval $[a,b]$ (right-continuous in $a$ , left-continuous in $b$ , all points between $a$ and $b$ ), and $M$ lies between the values of $f(a)$ and $f(b)$ , then there is at least one point $c$ between $a$ and $b$ such that $f(c)=M$ .

Limits