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Glacius
CalculusConcept reference

One-sided limits

A one-sided limit is the output value approached when the input approaches a target using only inputs on the specified side.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. On this graph, approaching x=2 from the left means following inputs smaller than 2
  3. A filled circle tells you the value at the exact input
  4. Left and right refer to input positions, not positive or negative outputs
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01On this graph, approaching x=2 from the left means following inputs smaller than 2#

On this graph, approaching x=2x=2 from the left means following inputs smaller than 2. That branch approaches height 3, even though the filled point has height 5.

Local piecewise graph of f. The left continuous straight segment runs from (1.2,2) to an open point at (2,3). The right continuous straight segment runs from an open point at (2,1) to (2.8,2). The function value at the target is the filled point (2,5). These segments specify every nearby input on their respective sides. The horizontal axis is x; the vertical axis is f(x).Local piecewise graph of f. The left continuous straight segment runs from (1.2,2) to an open point at (2,3). The right continuous straight segment runs from an open point at (2,1) to (2.8,2). The function value at the target is the filled point (2,5). These segments specify every nearby input on their respective sides. The horizontal axis is x; the vertical axis is f(x).
Figure 1Local piecewise graph of f. The left continuous straight segment runs from (1.2,2) to an open point at (2,3). The right continuous straight segment runs from an open point at (2,1) to (2.8,2). The function value at the target is the filled point (2,5). These segments specify every nearby input on their respective sides. The horizontal axis is x; the vertical axis is f(x).
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The superscript minus means inputs below the target. The superscript plus means inputs above it. Read the branch on that side; an open circle can mark the height it approaches.

x→a− : leftx→a+ : right\begin{gathered}x\to a^-\text{ : left}\\x\to a^+\text{ : right}\end{gathered}
Check your reasoning

Read lim⁡x→2−f(x)\lim_{x\to 2^{-}}f(x).

1352xf(x)0
Show answer and explanation
5

Left branch: 5.

02A filled circle tells you the value at the exact input#

A filled circle tells you the value at the exact input. A one-sided limit follows nearby inputs. Ignore a detached filled point when tracing the branch toward its endpoint.

Check your reasoning

Read lim⁡x→1+f(x)\lim_{x\to 1^{+}}f(x).

1351xf(x)0
Show answer and explanation
5

Right branch: 5.

03Left and right refer to input positions, not positive or negative outputs#

Left and right refer to input positions, not positive or negative outputs. The two sides may approach different heights. A filled endpoint on the chosen branch is also compatible with its one-sided limit.

Check your reasoning

Sensor: approach setting 3 from below. Limit?

1353xf(x)0
Show answer and explanation
1

Left branch: 1.

Key takeaway

Choose the input side, then follow its graph.

  • Determine a one-sided limit from a piecewise graph

Further questions

Must the left and right limits match?
No. Each follows only one branch. A jump can have two different finite one-sided limits. Matching them becomes relevant when asking for a two-sided limit.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “One-sided limits.” Math behind ML. /learn/c-one-sided