# What is the formal definition of limit?

Sep 17, 2014

Precise Definitions

Finite Limit
${\lim}_{x \to a} f \left(x\right) = L$ if
for all $\epsilon > 0$, there exists $\delta > 0$ such that
$0 < | x - a | < \delta R i g h t a r r o w | f \left(x\right) - L | < \epsilon$

Infinite Limits
${\lim}_{x \to a} f \left(x\right) = + \infty$ if
for all $M > 0$, there exists $\delta > 0$ such that
$0 < | x - a | < \delta R i g h t a r r o w f \left(x\right) > M$

${\lim}_{x \to a} f \left(x\right) = - \infty$ if
for all $N < 0$, there exists $\delta > 0$ such that
$0 < | x - a | < \delta R i g h t a r r o w f \left(x\right) < N$