Definition:Limit

From CS2800 wiki

If [math]f [/math] is a function from [math]\href{/cs2800/wiki/index.php?title=Real_number&action=edit&redlink=1}{\mathbb{R}} [/math] to [math]\href{/cs2800/wiki/index.php?title=Real_number&action=edit&redlink=1}{\mathbb{R}} [/math], we say that the limit of [math]f [/math] at [math]x_0 [/math] is [math]a [/math] (written [math]\href{/cs2800/wiki/index.php/%5Clim}{\lim}_{x → x_0} f = a [/math]) if for all tolerances [math]ε \gt 0 [/math], there exists a distance [math]δ \gt 0 [/math] such that for any value [math]x [/math] that is within the range [math](x_0 - δ, x_0 + δ) [/math], the distance between [math]f(x) [/math] and [math]a [/math] is smaller than the tolerance.

More concisely, [math]\href{/cs2800/wiki/index.php/%5Clim}{\lim}_{x → x_0} f = a [/math] means [math]\href{/cs2800/wiki/index.php/%E2%88%80}{∀} ε \gt 0, \href{/cs2800/wiki/index.php/%E2%88%83}{∃} δ \lt 0, \href{/cs2800/wiki/index.php/%E2%88%80}{∀} x, \text{ if } 0 \lt |x - x_0| \lt δ \text{ then } |f(x) - a| \lt ε [/math].