Right-Hand Limit

A function f(x), defined on an interval (c, d), is said to have a right-hand limit as x approaches c from the right if: $$ \lim_{x \rightarrow c^+} f(x) = l $$ In other words, for any ε > 0, there exists a δ > 0 such that whenever $$ c < x < c + δ, \text{ then } |f(x) - l| < ε. $$

For the right-hand limit to exist, this condition must hold over the open interval (c, c + δ), which includes values strictly greater than c.

Note. When evaluating the right-hand limit of a function at a point c, we consider only values approaching c from the right. Values to the left of c - or the value of the function exactly at c - are not relevant for the right-hand limit and may or may not exist independently.

A Practical Example

Let’s examine the function f(x) = x² at the point c = 2.

The right-hand limit of the function is:

$$ \lim_{x \rightarrow 2^+} f(x) = 4 $$

Verification

Let’s choose an arbitrary value ε = 5, which is greater than zero.

$$ l + ε = 4 + 5 = 9 $$

This allows us to select a value δ = 1, also greater than zero, and define the interval:

$$ (c, c + δ) = (2, 2 + 1) = (2, 3) $$

For any x within the open interval (2, 3) on the x-axis, the absolute difference between f(x) and the right-hand limit l is less than ε.

Therefore, the right-hand limit of the function f(x) at x = 2 is indeed 4.