Right-Hand and Left-Hand Limits of a Function

The limit of a function as x approaches x₀ can be evaluated either from the right (x → x₀⁺), meaning we approach x₀ with values greater than x₀. This is known as the right-hand limit:
$$ \lim_{x \rightarrow x_0^+} f(x) $$ or from the left (x → x₀^-), meaning we approach x₀ with values less than x₀. This is called the left-hand limit:
$$ \lim_{x \rightarrow x_0^-} f(x) $$

The right-hand or left-hand limit may be finite or infinite.

At certain points, a function may not possess either a right-hand or left-hand limit.

Example. The logarithmic function f(x) = log(x) is defined only for positive real numbers, i.e. x ∈ (0, +∞). As a result, the limit as x approaches 0 exists solely from the right, that is, as x → 0⁺. There is no left-hand limit as x approaches zero.

Right-Hand Limit of a Function

Definition of the right-hand limit:

The right-hand limit of a function f(x) as x → x₀⁺ equals l if, for every ε > 0, there exists a δ > 0 such that |f(x) - l| < ε for all x satisfying x₀ < x < x₀ + δ:
$$ \lim_{x \rightarrow x_0^+} f(x) = l $$

Expressed symbolically:

$$ \lim_{x \rightarrow x_0^+} f(x) = l \Leftrightarrow \forall \epsilon>0, \exists \delta>0: |f(x)-l|<\epsilon, \forall x \in (x_0, x_0+\delta) $$

Note. The right-hand limit can also be described using sequences: if we consider a sequence x_n approaching x₀ from the right, then f(x_n) converges to l as n grows large.

Example

The function f(x) = 1/x is undefined at x = 0:

$$ f(x) = \frac{1}{x} $$

We evaluate the limit as x → 0⁺ from the right without ever reaching x = 0:

$$ \lim_{x \rightarrow 0^+} \frac{1}{x} $$

The right-hand limit exists and equals +∞:

$$ \lim_{x \rightarrow 0^+} \frac{1}{x} = +\infty $$

Here’s how this appears on the graph:

Left-Hand Limit of a Function

Definition of the left-hand limit:

The left-hand limit of a function f(x) as x → x₀^- equals l if, for every ε > 0, there exists a δ > 0 such that |f(x) - l| < ε for all x satisfying x₀ - δ < x < x₀:
$$ \lim_{x \rightarrow x_0^-} f(x) = l $$

Expressed symbolically:

$$ \lim_{x \rightarrow x_0^-} f(x) = l \Leftrightarrow \forall \epsilon>0, \exists \delta>0: |f(x)-l|<\epsilon, \forall x \in (x_0-\delta, x_0) $$

Note. The left-hand limit can also be defined using sequences: if we take a sequence x_n approaching x₀ from the left, then f(x_n) approaches l as n increases.

Example

The function f(x) = 1/x is undefined at x = 0:

$$ f(x) = \frac{1}{x} $$

We evaluate the limit as x → 0^- from the left without ever reaching x = 0:

$$ \lim_{x \rightarrow 0^-} \frac{1}{x} $$

The left-hand limit exists and equals -∞:

$$ \lim_{x \rightarrow 0^-} \frac{1}{x} = -\infty $$

Note. In this case, the limit approaches negative infinity because x approaches zero from the left. Thus, the values of x remain negative, such as x = -0.3, x = -0.1, x = -0.01, and so on.

Here’s how this is represented on the Cartesian plane:

And so on.