One-sided limit - Ciallo Wiki

1 Definition and directional approach

A one-sided limit describes how a function behaves as the input variable approaches a specified point while remaining on only one side of it. Concretely, for a point \(a\) and a function \(f\), the left-hand limit examines \(x \to a^-\) (values less than \(a\)), while the right-hand limit examines \(x \to a^+\) (values greater than \(a\)). In both cases, the function values are compared along approaches to \(a\) that stay within the chosen direction, without requiring information from the opposite side.

1.1 Left-hand limit

The left-hand limit of \(f(x)\) as \(x\) approaches \(a\) is denoted \[ \lim_{x\to a^-} f(x) = L, \] meaning that values of \(f(x)\) become arbitrarily close to \(L\) when \(x\) is sufficiently near \(a\) from the left.

1.1.1 Formal \(\varepsilon\)-\(\delta\) characterization

\[ \lim_{x\to a^-} f(x) = L \] if and only if for every \(\varepsilon>0\) there exists \(\delta>0\) such that whenever \[ 0 < a-x < \delta, \] it follows that \[

f(x)-L	< \varepsilon.

\] The inequality \(0<a-x\) enforces that \(x\) remains to the left of \(a\).

1.1.2 Sequence-based criterion

An equivalent way to express a one-sided limit uses sequences. One has \[ \lim_{x\to a^-} f(x)=L \] if and only if for every sequence \((x_n)\) with \(x_n<a\) and \(x_n\to a\), the sequence \((f(x_n))\) satisfies \(f(x_n)\to L\). This reformulates the “approach from the left” idea in a purely sequential manner.

1.2 Right-hand limit

The right-hand limit of \(f(x)\) as \(x\) approaches \(a\) is denoted \[ \lim_{x\to a^+} f(x) = R, \] indicating that \(f(x)\) approaches \(R\) when \(x\) approaches \(a\) from values greater than \(a\).

1.2.1 Formal \(\varepsilon\)-\(\delta\) characterization

\[ \lim_{x\to a^+} f(x)=R \] if and only if for every \(\varepsilon>0\) there exists \(\delta>0\) such that whenever \[ 0 < x-a < \delta, \] we have \[

f(x)-R	<\varepsilon.

\] Here \(0<x-a\) ensures \(x\) stays to the right of \(a\).

1.2.2 Sequence-based criterion

Equivalently, \[ \lim_{x\to a^+} f(x)=R \] if and only if for every sequence \((x_n)\) with \(x_n>a\) and \(x_n\to a\), the values satisfy \(f(x_n)\to R\).

1.3 Existence of one-sided limits

A one-sided limit may exist even when the function behaves differently on the other side of the point.

1.3.1 When a two-sided limit exists

If both one-sided limits exist and are equal—i.e., \[ \lim_{x\to a^-} f(x)=L \quad \text{and} \quad \lim_{x\to a^+} f(x)=L, \] then the two-sided limit exists and equals \(L\). In that case, approaching \(a\) from either direction produces the same limiting behavior.

1.3.2 When one side exists but the other does not

It is also possible that only one one-sided limit exists. For example, the left-hand limit might approach a finite value while the right-hand values oscillate without settling, or diverge to infinity. In such circumstances, the two-sided limit fails to exist, but one-sided information remains valid and useful—for instance, in continuity and differentiability criteria at boundary points.

2 Relationship to two-sided limits

Two-sided limits combine directional behavior. The comparison between one-sided limits and the overall two-sided limit is central to diagnosing continuity and classifying discontinuities.

2.1 Equality conditions

The key role of one-sided limits is captured by precise equality requirements.

2.1.1 Matching left and right limits

If both one-sided limits exist and satisfy \[ \lim_{x\to a^-} f(x)=\lim_{x\to a^+} f(x)=M, \] then \[ \lim_{x\to a} f(x)=M. \] Thus, the two-sided limit exists exactly when the left-hand and right-hand limits not only exist but also agree.

2.1.2 Consequences for the two-sided limit

If the one-sided limits differ—say one equals \(L\) and the other equals \(R\) with \(L\neq R\)—then the two-sided limit cannot exist. Similarly, if one one-sided limit exists and the other fails to exist (oscillatory or divergent behavior), the two-sided limit does not exist either. The two-sided limit is therefore a stricter requirement than either one-sided condition alone.

2.2 Common failure patterns

Many functions fail to have a two-sided limit because their left and right behaviors do not match.

2.2.1 Jump discontinuity

A typical example of mismatch is a jump discontinuity, where one-sided limits exist but take different finite values: \[ \lim_{x\to a^-} f(x)=L,\quad \lim_{x\to a^+} f(x)=R,\quad L\neq R. \] Graphically, the function approaches different heights depending on direction.

2.2.2 Infinite one-sided behavior

Another source of failure is divergence. One side may increase without bound while the other remains finite, or both may diverge with different signs. In such cases, the two-sided limit may fail because at least one directional approach does not converge to a finite value (though infinite limits can be treated in an extended sense).

3 Computation techniques

Calculating one-sided limits often reduces to algebraic simplification, valid substitutions, and standard limit laws, each applied under appropriate assumptions.

3.1 Algebraic limit evaluation

3.1.1 Direct substitution when valid

When \(f\) is defined by a formula that is well-behaved from the relevant side, one may sometimes compute the one-sided limit by substituting \(x=a\) into the expression and then interpreting the result as the limiting value. This is valid when the expression has no problematic behavior as \(x\) approaches \(a\) from that side.

3.1.2 Simplification near the point

When substitution is not valid (for instance, due to a removable factor or an indeterminate form), algebraic manipulation is used. Common approaches include factoring, canceling common terms, or rewriting expressions so that the behavior near \(a\) becomes explicit. For one-sided limits, it is particularly important not to assume cancellation is legitimate if it depends on crossing through problematic values that only affect one direction.

3.2 Limit laws for one-sided limits

Limit laws extend to one-sided limits with suitable conditions.

3.2.1 Sum, product, and quotient rules

If both relevant one-sided limits exist (and for quotients, the denominator limit is nonzero), then one can apply:

\(\lim_{x\to a^\pm}(f+g)=\lim_{x\to a^\pm}f+\lim_{x\to a^\pm}g\),
\(\lim_{x\to a^\pm}(fg)=\left(\lim_{x\to a^\pm}f\right)\left(\lim_{x\to a^\pm}g\right)\),
\(\lim_{x\to a^\pm}\left(\frac{f}{g}\right)=\frac{\lim_{x\to a^\pm}f}{\lim_{x\to a^\pm}g}\), when \(\lim_{x\to a^\pm}g\neq 0\).

These rules reflect that the “approach” is restricted to one side, but the limiting algebra proceeds as in the two-sided setting.

3.2.2 Composition with continuous functions

If \(\phi\) is continuous at the limiting value and \(\lim_{x\to a^\pm} f(x)=L\), then \[ \lim_{x\to a^\pm} \phi(f(x))=\phi(L). \] Continuity of \(\phi\) ensures that small changes in \(f(x)\) produce small changes in \(\phi(f(x))\), preserving the limiting behavior.

3.3 Squeezing arguments

3.3.1 One-sided squeeze theorem

A one-sided version of the squeeze theorem applies when two bounding functions squeeze the target function as \(x\to a^\pm\). If for \(x\) sufficiently near \(a\) from the chosen side, \[ g(x)\le f(x)\le h(x), \] and \[ \lim_{x\to a^\pm} g(x)=\lim_{x\to a^\pm} h(x)=M, \] then \[ \lim_{x\to a^\pm} f(x)=M. \] The proof mirrors the two-sided theorem, but all inequalities and neighborhoods are taken only on the specified side.

3.3.2 Comparison via bounding sequences

Sometimes bounding directly by functions is inconvenient, while bounding along sequences from the relevant side is feasible. If every sequence \(x_n\to a\) from the chosen side forces \(f(x_n)\) to be trapped between sequences with the same limit, then \(f(x_n)\) must share that limit. This provides a sequence-based computation strategy consistent with the limit definition.

4 Graphical and geometric interpretation

Graphs encode one-sided limits through how the curve approaches a point from left or right.

4.1 Reading one-sided limits from graphs

4.1.1 Left and right approach values

To read a left-hand limit from a graph, trace the function values as \(x\) approaches the point \(a\) using only points with \(x<a\). The left-hand limit equals the height the graph approaches. Likewise, the right-hand limit comes from the behavior using \(x>a\).

4.1.2 Filled vs open circles and continuity

On a graph, an open circle at \((a,y)\) indicates the function does not take that value, but the curve may still approach it. A filled circle indicates the function value at \(x=a\). One-sided limits depend on nearby values, so the presence or absence of a filled point at \(x=a\) affects continuity but not the limit itself, provided the surrounding curve approaches a definite height.

4.2 Vertical asymptotes from one side

4.2.1 \(-\infty\) and \(+\infty\) one-sided divergence

If as \(x\to a^-\) the function decreases without bound, then \(\lim_{x\to a^-} f(x)=-\infty\). If it increases without bound from the left, the limit is \(+\infty\). An analogous interpretation holds for \(x\to a^+\). Different signs on the two sides often appear as vertical asymptotes with opposite directional behavior.

4.2.2 Rate-of-blow-up intuition

Although the formal one-sided limit records only the presence of divergence (finite value or infinity), the graph’s steepness can suggest how quickly the magnitude grows. This intuition is helpful for qualitative understanding, but rigorous classification still relies on the limiting definition rather than visual “speed.”

5 Discontinuity and classification

One-sided limits are the primary tools for describing and classifying discontinuities at a point.

5.1 Relation between one-sided limits and discontinuity

5.1.1 Jump discontinuity (finite limits)

At a point \(a\), a jump discontinuity occurs when both one-sided limits exist and are finite but unequal: \[ \lim_{x\to a^-} f(x)=L,\quad \lim_{x\to a^+} f(x)=R,\quad L\neq R. \] Here the function fails to approach a single consistent value as the input approaches from both sides.

5.1.2 Removable discontinuity vs one-sided limits

A removable discontinuity often arises when the limit \(\lim_{x\to a} f(x)\) exists but the function value \(f(a)\) is undefined or different from the limiting value. In this situation, the one-sided limits match: \[ \lim_{x\to a^-} f(x)=\lim_{x\to a^+} f(x)=M, \] yet \(f(a)\neq M\) or \(f(a)\) does not exist. The mismatch is therefore between the function’s value at the point and its limiting behavior.

5.2 Infinite discontinuities

5.2.1 Divergence from one side

Infinite discontinuities occur when the function does not settle to a finite value as \(x\to a\) from at least one side. For example, if \(\lim_{x\to a^+} f(x)=+\infty\), then the function blows up when approached from the right, typically indicating a vertical asymptote on that side.

5.2.2 Mixed finite/infinite behavior

A point can show mixed behavior: one side may converge to a finite number while the other diverges. Such a configuration prevents the two-sided limit from existing and indicates a more severe type of discontinuity than a mere jump between finite values.

6 Continuity at and near points of a domain

Continuity connects the limiting process to the function’s actual value, with one-sided limits providing the correct notion at boundary points.

6.1 Continuity on open intervals

6.1.1 Two-sided continuity criterion

A function \(f\) is continuous at an interior point \(a\) of its domain if:

\(f(a)\) is defined, and
\(\lim_{x\to a} f(x)=f(a)\).

Because \(a\) is interior, this uses the standard two-sided limit, equivalent to requiring both one-sided limits to exist and equal \(f(a)\).

6.2 Continuity at endpoints

6.2.1 One-sided continuity definition

If \(a\) is a left endpoint of the domain, there is no need to consider \(x<a\); continuity is defined using the right-hand limit: \[ f(a) \text{ is continuous at } a \quad \Longleftrightarrow \quad \lim_{x\to a^+} f(x)=f(a). \] If \(a\) is a right endpoint, continuity uses \(\lim_{x\to a^-}\) instead. This ensures the definition respects the available approach directions within the domain.

6.2.2 Examples on half-open domains

On a half-open interval such as \([0,1)\), continuity at \(x=0\) is evaluated using right-hand behavior, while continuity at points \(0<x<1\) uses two-sided limits. The one-sided framework prevents incorrectly demanding approaches that are not part of the domain.

7 Differentiability and higher-order behavior

Differentiability is defined using difference quotients, and one-sided versions arise naturally at points where only one direction is available.

7.1 One-sided derivatives

7.1.1 Definition using one-sided difference quotients

The right-hand derivative at \(a\) is defined (when the limit exists) by \[ f'_+(a)=\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h}. \] Similarly, the left-hand derivative is \[ f'_-(a)=\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h}. \] These definitions use increments \(h\) restricted to one sign, matching the directional approach.

7.1.2 Existence and meaning

A one-sided derivative measures the slope of the function’s graph as it is approached from the corresponding side. If \(f'_+(a)\) exists but \(f'_-(a)\) does not, the function has a well-defined “approach slope” only from one direction. When both one-sided derivatives exist, their equality determines whether the derivative in the usual two-sided sense exists.

7.2 Links to one-sided limits of functions

7.2.1 Derivative implies continuity (where applicable)

Where \(f'\) exists in a one-sided sense and \(f(a)\) is defined, the function must be continuous at \(a\) from that side. Intuitively, if the graph has a stable directional slope and the difference quotient converges, the function values cannot jump along that approach. (More precisely, one-sided differentiability entails one-sided continuity at the relevant boundary points.)

7.2.2 Corner/cusp behavior via one-sided slopes

At a corner or cusp, one-sided derivatives often exist but differ. For instance, a function shaped like a “V” has distinct left and right slopes at the vertex; this produces a derivative that fails to exist in the two-sided sense even though directional slopes are well-defined. Such behavior is detected by comparing \(f'_-(a)\) and \(f'_+(a)\).

8 Worked examples and typical test cases

Example patterns illustrate how one-sided limits behave in common analytic tasks.

8.1 Piecewise functions

8.1.1 Limits at partition points

For a piecewise function defined by different formulas on either side of \(a\), one-sided limits are typically computed by substituting the appropriate branch and taking the limit within that branch’s region.

A common test case uses a definition like:

\(f(x)=g(x)\) for \(x<a\),
\(f(x)=h(x)\) for \(x>a\).

Then: \[ \lim_{x\to a^-} f(x)=\lim_{x\to a^-} g(x), \quad \lim_{x\to a^+} f(x)=\lim_{x\to a^+} h(x). \] The two-sided limit exists only if these values match.

8.2 Functions with absolute value and roots

8.2.1 Handling symmetric vs asymmetric approach

Absolute value functions frequently behave differently depending on whether the input is above or below the point. When computing one-sided limits, the absolute value may resolve differently on each side, producing distinct limiting values. Roots can similarly create asymmetric behavior near a point if the expression under the radical forces a restricted real domain (or if the inside approaches zero differently on different sides).

8.3 Trigonometric and rational expressions near a point

8.3.1 Approach-dependent simplification

Trig expressions like \(\sin\) and \(\cos\) are continuous everywhere, so discontinuities typically come from compositions involving discontinuous algebraic factors. In one-sided computations, the key is whether the potentially problematic factor (such as a denominator) stays nonzero or approaches zero from only one side.

8.3.2 Poles and removable singularities

Rational functions often create one-sided issues at points where the denominator vanishes. If cancellation is possible, the singularity can be removable, leading to matching one-sided limits. If not, one-sided divergence may occur, sometimes with different signs depending on the direction of approach.

9 Common pitfalls and clarifications

Many errors in one-sided limits stem from mixing up directional requirements, conflating the function value with the limiting value, or using limit laws without checking conditions.

9.1 Confusing function value with limit

The limit concerns behavior near the point, not necessarily the value at the point. A function may have a perfectly good one-sided limit while \(f(a)\) is undefined or differs from the limit.

9.2 Mistaking domain restrictions for one-sided limits

A domain restriction may mean the expression is not meaningful on one side of \(a\). It does not automatically justify assuming a one-sided limit exists; it only indicates which approaches are allowed. One must still verify convergence along the permitted direction.

9.3 Assuming existence of a two-sided limit from one side only

A one-sided limit existing does not imply the two-sided limit exists. The opposite direction might diverge or oscillate, preventing a common limiting value.

9.4 Misapplying limit laws without verifying one-sided conditions

Limit laws require the relevant one-sided limits to exist and satisfy the needed constraints (e.g., nonzero denominators for quotient rules). Using algebraic manipulations without ensuring the hypotheses on the correct side can lead to incorrect conclusions.