Formally, the limit of a function, f(x), as x approaches a number p, is equal to L only if for every epsilon greater than zero (ε > 0), there exists a delta greater than zero (δ > 0), such that for all real x, 0 < | x ' p | < δ implies | f(x) ' L | < ε. In other words, as the distance between x and p gets very small, the distance between f(x) and L also gets very small.
Limits can be taken in two directions. For example, if p is equal to zero, the function, f(x) can approach p from the right, in which the values close to p will be positive. Conversely, f(x) can approach p from the left and its values will be negative. The limit of a function can be shown to equal L only if its left- and right-hand limits are equal.
Take the limit of sin x as x goes to zero from the right. We need to show for positive x very close to zero, | sin x - 0 | < ε. In other words, | sin x | < ε. But since sin x < x for small positive x, we can let δ = ε and find for x < δ, sin x < ε. This completes the proof for the right-hand side. The left-hand limit is found by the same method, with the absolute values removing the negative signs.
Because the left-hand and right-hand limits of sin x are both equal to zero as x approaches zero, the limit of sin x as x approaches zero is zero. This limit appears often in math, as in the more complicated limit of sin x/x as x approaches zero. This limit is challenging, but a proof can show it equals 1.