Limit

1. of a function

The value \(y\), if it exists, that a function \(f\) takes as its argument \(x\) approaches some value \(c\), written as
\[ \lim_{x \to c} f(x) = y \]
or
\[ f(x) \underset{x \to c}{=} y \]
Note that the limit may have a value even if the terms of the function seem to imply that it doesn't. For example
\[ \lim_{x \to 0} \frac{\sin x} {x} = 1 \]
even though \(\sin 0 = 0\).

2. of a sequence

The value \(y\), if it exists, that the terms of a sequence \(x_i\) approach as \(i\) increases, written as
\[ \lim_{i \to \infty} x_i = y \]
or
\[ x_i \underset{i \to \infty}{=} y \]