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One-sided limit

One-sided limit is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand One-sided limit rather than just read about it. In short: In calculus, a one-sided limit refers to either one of the two limits of a function f ( x ) {\displaystyle f(x)} of a real variable x {\displaystyle x} as x {\displaystyle x} approaches a specified point either from the left or from the right. The limit, as x {\displaystyle x} decreases in value approaching a {\displaystyle a} ( x {\displaystyle x} approaches a {\displaystyle a} "from the right" or "from above"), is…

One-sided limit — main illustration
One-sided limit — illustration

Key takeaways

  • One-sided limit belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect One-sided limit to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of One-sided limit from memory before moving on to harder problems.

Reference excerpt

In calculus, a one-sided limit refers to either one of the two limits of a function f ( x ) {\displaystyle f(x)} of a real variable x {\displaystyle x} as x {\displaystyle x} approaches a specified point either from the left or from the right. The limit, as x {\displaystyle x} decreases in value approaching a {\displaystyle a} ( x {\displaystyle x} approaches a {\displaystyle a} "from the right" or "from above"), is denoted:

lim x → a + f ( x ) or lim x ↓ a f ( x ) or lim x ↘ a f ( x ) or f ( a + ) . {\displaystyle \lim _{x\to a^{+}}f(x)\quad {\text{ or }}\quad \lim _{x\,\downarrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\searrow a}\,f(x)\quad {\text{ or }}\quad f(a+).}

The limit, as x {\displaystyle x} increases in value approaching a {\displaystyle a} ( x {\displaystyle x} approaches a {\displaystyle a} "from the left" or "from below"), is denoted:

lim x → a − f ( x ) or lim x ↑ a f ( x ) or lim x ↗ a f ( x ) or f ( a − ) . {\displaystyle \lim _{x\to a^{-}}f(x)\quad {\text{ or }}\quad \lim _{x\,\uparrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\nearrow a}\,f(x)\quad {\text{ or }}\quad f(a-).}

If the limits from the left and right both exist and are equal, then the limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches a {\displaystyle a} exists. Conversely, if the limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches a {\displaystyle a} exists, then the limits from left and right both exist and are equal. Consequently, the limit as x {\displaystyle x} approaches a {\displaystyle a} is sometimes called a "two-sided limit". It is denoted:

lim x → a f ( x ) . {\displaystyle \lim _{x\to a}f(x).}

In some cases in which the two-sided limit does not exist, the two individual one-sided limits nonetheless exist and they are then necessarily unequal. It is possible for only one of the two one-sided limits to exist. It is also possible for neither of the two one-sided limits to exist.

Formal definition

Definition If I {\displaystyle I} represents some interval that is contained in the domain of a function f {\displaystyle f} and if a {\displaystyle a} is a point in I {\displaystyle I} , then the right-sided limit as x {\displaystyle x} approaches a {\displaystyle a} can be rigorously defined as the value R {\displaystyle R} that satisfies:

for all ε > 0 {\displaystyle \varepsilon >0} there exists some δ > 0 {\displaystyle \delta >0} such that for all x ∈ I {\displaystyle x\in I} , if 0 < x − a < δ {\displaystyle 0<x-a<\delta } then | f ( x ) − R | < ε {\displaystyle |f(x)-R|<\varepsilon } , and the left-sided limit as x {\displaystyle x} approaches a {\displaystyle a} can be rigorously defined as the value L {\displaystyle L} that satisfies:

… excerpt ends here. Continue reading the full article.

Illustrations

One-sided limit: At x = 0 , {\displaystyle x=0,} the function f ( x ) = x 2 + sign ⁡ ( x ) , {\displaystyle f(x)=x^{2}+\operatorname {sign} (x),} where sign ⁡ ( x ) {\displaystyle \operatorname {sign} (x)} denotes the sign function, has a left limit of − 1 , {\displaystyle -1,} a right limit of + 1 , {\displaystyle +1,} and a function value of 0. {\displaystyle 0.}
At x = 0 , {\displaystyle x=0,} the function f ( x ) = x 2 + sign ⁡ ( x ) , {\displaystyle f(x)=x^{2}+\operatorname {sign} (x),} where sign ⁡ ( x ) {\displaystyle \operatorname {sign} (x)} denotes the sign function, has a left limit of − 1 , {\displaystyle -1,} a right limit of + 1 , {\displaystyle +1,} and a function value of 0. {\displaystyle 0.}
One-sided limit: Plot of the function f ( x ) = 1 1 + 2 − 1 / x {\textstyle f(x)={\frac {1}{1+2^{-1/x}}}} .
Plot of the function f ( x ) = 1 1 + 2 − 1 / x {\textstyle f(x)={\frac {1}{1+2^{-1/x}}}} .

Worked examples

Example 1 — a first encounter with One-sided limit

Start with the simplest possible case. Write down what One-sided limit claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to One-sided limit before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about One-sided limit ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of One-sided limit

In research
One-sided limit appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses One-sided limit in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
One-sided limit is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functions and mappings, Limits (mathematics), Real analysis, so understanding it makes those chapters shorter.
In everyday life
Look for One-sided limit outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study One-sided limit in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what One-sided limit means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain One-sided limit out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is One-sided limit in simple terms?

In calculus, a one-sided limit refers to either one of the two limits of a function f ( x ) {\displaystyle f(x)} of a real variable x {\displaystyle x} as x {\displaystyle x} approaches a specified point either from the left or from the right. The limit, as x {\displaystyle x} decreases in value ap…

Why does One-sided limit matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study One-sided limit?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on One-sided limit.

Tags

  • Functions and mappings
  • Limits (mathematics)
  • Real analysis

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