Left/Right-Hand Limits and Two-Sided Limits

Key idea: To analyze limits at a point a, inspect the behavior of f(x) as x approaches a from the left side (values less than a).
Notation:
- Left-hand limit exists if there is a number L such that $\lim_{x \to a^-} f(x) = L.$
Formal definition (epsilon–delta):
- For a given a, if \forall \varepsilon > 0\,\exists \delta > 0\text{ such that } 0 < a - x < \delta \Rightarrow |f(x) - L| < \varepsilon, then $\lim_{x \to a^-} f(x) = L.$
- Intuition: as x approaches a from values smaller than a, f(x) gets arbitrarily close to L.
Domain requirement: a must be a limit point of the domain of f from the left (i.e., there are x in the domain with x < a arbitrarily close to a).
Examples:
- If f(x) = x² and a = 2, then $\lim_{x \to 2^-} f(x) = 4.$
- If f(x) = |x| and a = 0, then $\lim_{x \to 0^-} f(x) = 0.$
Endpoints: when a is the left endpoint of the domain (e.g., domain is x ≥ a), the left-hand limit may not exist; in such cases only the right-hand limit is typically meaningful.
Significance: left-hand limits help diagnose one-sided behavior and are essential for piecewise definitions and discontinuities on the left side.

Key idea: To analyze limits at a point a, inspect the behavior of f(x) as x approaches a from the right side (values greater than a).
Notation:
- Right-hand limit exists if there is a number M such that $\lim_{x \to a^+} f(x) = M.$
Formal definition (epsilon–delta):
- For a given a, if \forall \varepsilon > 0\,\exists \delta > 0\text{ such that } 0 < x - a < \delta \Rightarrow |f(x) - M| < \varepsilon, then $\lim_{x \to a^+} f(x) = M.$
Domain requirement: a must be a limit point of the domain from the right (i.e., there are x in the domain with x > a arbitrarily close to a).
Examples:
- If f(x) = x² and a = 2, then $\lim_{x \to 2^+} f(x) = 4.$
- If f(x) = 1/x and a = 0, the right-hand limit does not exist as a finite value (it tends to +∞ or −∞ depending on direction).
Endpoints: when a is the right endpoint of the domain (e.g., domain is x ≤ a), the right-hand limit may not exist; only the left-hand limit is typically meaningful.
Significance: right-hand limits reveal behavior immediately to the right of a and are crucial for one-sided continuity checks and piecewise definitions.

Key idea: The two-sided limit exists only if both one-sided limits exist and are equal.
Notation:
- If both exist and are equal to L, then $\lim_{x \to a} f(x) = L.$
Formal condition:
- $\lim{x \to a} f(x) = L \quad\text{iff}\quad \lim{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.$
If the left-hand and right-hand limits exist but are different, the two-sided limit does not exist.
If one or both one-sided limits do not exist (or are infinite), the two-sided limit does not exist (as a finite value).
Connection to continuity:
- A function is continuous at a point a if and only if $\lim_{x \to a} f(x) = f(a).$
- Even if the function value f(a) is defined, a mismatch between the limit and f(a) indicates a discontinuity at a.
Examples:
- f(x) = x², a = 2: left = right = 4 ⇒ $\lim_{x \to 2} f(x) = 4.$ If f(2) = 5, the function is discontinuous at 2 even though the limit exists.
- Heaviside step function H(x) defined by $H(x) = \begin{cases}0, & x < 0 \\ 1, & x \ge 0\end{cases}$
- $\lim{x \to 0^-} H(x) = 0,$ $\lim{x \to 0^+} H(x) = 1.$ The two-sided limit at 0 does not exist.
- f(x) = \frac{1}{x} near a = 0:
- Left-hand limit does not exist as a finite number (tends to $-\infty$), right-hand limit tends to $+\infty$; two-sided limit does not exist as a finite value.

Always check both sides when a is an interior point of the domain.
If a is a boundary point of the domain, only one-sided limit may be applicable.
Even if the limit exists, the function may be undefined at a or have a different value (discontinuity of the first kind vs second kind):
- If $\lim_{x \to a} f(x) = L$ and f(a) ≠ L, the function is discontinuous at a (remains a finite limit).
Common discontinuities:
- Jump discontinuity: left-hand limit and right-hand limit exist but are unequal (e.g., Heaviside function at 0).
- Removable discontinuity: left and right limits exist and are equal to L, but f(a) ≠ L or f(a) is undefined.
- Infinite discontinuity: one-sided limits go to ±∞.

Example 1: Evaluate left, right, and two-sided limits
- f(x) = (x^2 - 1)/(x - 1), a = 1
- For x ≠ 1, simplify: $f(x) = x + 1.$
- $\lim{x \to 1^-} f(x) = 2,$ $\lim{x \to 1^+} f(x) = 2.$
- Since both one-sided limits exist and are equal, $\lim_{x \to 1} f(x) = 2.$
- If one defines f(1) = 3, the function is still continuous at 1 because the limit equals f(1) only if f(1) = 2; otherwise point discontinuity.
Example 2: Jump discontinuity
- H(x) = $\begin{cases}0, & x < 0 \\ 1, & x \ge 0\end{cases}$
- Left limit at a = 0: $\lim{x \to 0^-} H(x) = 0.$ Right limit: $\lim{x \to 0^+} H(x) = 1.$ Two-sided limit does not exist.
Example 3: Infinite behavior
- f(x) = \frac{1}{x} at a = 0
- Left-hand limit: $\lim{x \to 0^-} f(x) = -\infty.$ Right-hand limit: $\lim{x \to 0^+} f(x) = +\infty.$ Two-sided limit does not exist as a finite value.

In modeling and analysis, knowing one-sided limits helps describe behavior at boundaries and in piecewise definitions.
Continuity tests rely on the two-sided limit; even if a function is defined at a, matching the limit ensures no jump.
In numerical methods, approximating limits near discontinuities requires awareness of one-sided behavior to avoid misleading results.
End behavior and asymptotics can be studied via one-sided limits at points where the function behaves differently on each side.

Problem 1: Determine the one-sided and two-sided limits if possible:
- f(x) = x^2 - 3x, a = 2
- f(x) = H(x) as defined above, a = 0
- f(x) = \frac{1}{x - 1}, a = 1
Problem 2: For each function, determine whether the function is continuous at the given point:
- f(x) = \begin{cases} x, & x \neq 2 \\ 5, & x = 2 \end{cases}, a = 2
- f(x) = |x|, a = 0
- f(x) = \frac{\sin x}{x}, a = 0 (define f(0) = 1 for the purposes of continuity)

Left-hand limit: $\lim_{x \to a^-} f(x) = L.$
Right-hand limit: $\lim_{x \to a^+} f(x) = M.$
Two-sided limit exists iff both one-sided limits exist and are equal: $\lim{x \to a} f(x) = L \text{ if } \lim{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L.$
Continuity at a requires the two-sided limit to equal the function value: $\lim_{x \to a} f(x) = f(a).$