Limits

$\lim_{x\to{a}}{f(x)}=L$ iff: $$

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (0<|x-a|<\delta\implies{|f(x)-L|<\epsilon})$$

Defining $\delta$ in terms of a given $\epsilon$ is enough to prove a limit.

Properties

Suppose $\lim f(x) = L, \lim g(x) =M$. $$

• $\lim {f \pm g} = L \pm M$
• $\lim {fg} = LM$
• $\lim \frac{f}{g} = \frac{L}{M}$ (provided $g(x)\neq 0$ and $M\neq 0$)

One sided limits

In x-limit

$$\lim_{x\to{a}}{f(x)}=L \iff \Big( \lim_{x\to{a^-}}{f(x)}=L \land \lim_{x\to{a^+}}{f(x)}=L \Big)$$

Right limit

$\lim_{x\to{a^{-}}}{f(x)}=L$ iff: $$

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (-\delta<x-a<0\implies{|f(x)-L|<\epsilon})$$

Left limit

$\lim_{x\to{a^{+}}}{f(x)}=L$ iff: $$

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (0<x-a<\delta\implies{|f(x)-L|<\epsilon})$$

In the answer

$$\lim_{x\to{a}}{f(x)}=L \iff \Big( \lim_{x\to{a}}{f(x)}=L^{+} \lor \lim_{x\to{a}}{f(x)}=L^{-} \Big)$$

Top limit

$\lim_{x\to{a}}{f(x)}=L^{+}$ iff: $$

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (0<\lvert{x-a}\rvert<\delta\implies{0\le f(x)-L<\epsilon})$$

Bottom limit

$\lim_{x\to{a}}{f(x)}=L^{-}$ iff: $$

$$\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (0<\lvert{x-a}\rvert<\delta\implies{-\epsilon\lt f(x)-L\le 0})$$

Limits including infinite

In x-limit

Positive infinity

$\lim_{x\to{\infty}}{f(x)}=L$ iff: $$

$$\forall{\epsilon\gt 0}\; \exists{N>0}\; \forall{x}\; (x\gt N\implies{|f(x)-L|<\epsilon})$$

Negative infinity

$\lim_{x\to{-\infty}}{f(x)}=L$ iff: $$

$$\forall{\epsilon\gt 0}\; \exists{N>0}\; \forall{x}\; (x\lt-N\implies{|f(x)-L|<\epsilon})$$

In the answer

Positive infinity

$\lim_{x\to a}{f(x)}=\infty$ iff: $$

$$\forall{M\gt 0}\; \exists{\delta>0}\; \forall{x}\; (0<\lvert{x-a}\rvert<\delta\implies{f(x)\gt M})$$

Negative infinity

$\lim_{x\to a}{f(x)}=-\infty$ iff: $$

$$\forall{M\gt 0}\; \exists{\delta>0}\; \forall{x}\; (0<\lvert{x-a}\rvert<\delta\implies{f(x)\lt-M})$$