limit

The value a function's outputs approach as the input approaches some point, without necessarily reaching it.

We write $\displaystyle\lim_{x\to a} f(x) = L$ when $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to (but not equal to) $a$ . The limit describes behavior near $a$ ; it says nothing about $f(a)$ itself, which may differ or not even exist.

Examples

$\displaystyle\lim_{x\to 2}(3x+1) = 7$ , and this holds even if $f(2)$ were redefined to be something else.