Left-Hand Limit

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

For the left-hand limit to exist, the definition must hold over the open interval (c - δ, c), which includes values strictly less than c.

Note. When evaluating the left-hand limit of a function at a point c, we only consider values approaching c from the left. We do not take into account values to the right of c (which pertain to the right-hand limit), nor the function’s value exactly at c itself. These may or may not exist independently of the left-hand limit.

A Practical Example

Let’s examine the function f(x) = x² at the point c = 3.

The left-hand limit of the function is:

$$ \lim_{x \rightarrow 3^-} f(x) = 9 $$

Verification

First, choose an arbitrary value ε = 5, which is greater than zero.

$$ l - ε = 9 - 5 = 4 $$

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

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

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

Therefore, the left-hand limit of the function f(x) at the point c is indeed 9.