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Limit from the Left — Definition, Formula & Examples

Limit from the Left
Limit from Below

A one-sided limit which, in the example lim⁡x→0−1x=−∞,\displaystyle \lim_{x \to 0^-}\frac{1}{x} = -\infty\text{,} restricts x such that x < 0.

In general, a limit from the left restricts the domain variable to values less than the number the domain variable approaches. When a limit is taken from the left it is written lim⁡x→a−f(x)\displaystyle \lim_{x \to a^-} f(x) or lim⁡x↑af(x).\displaystyle \lim_{x \uparrow a} f(x)\text{.}

For example, lim⁡x→0−1x=−∞\displaystyle \lim_{x \to 0^-}\frac{1}{x} = -\infty since 1x\displaystyle \frac{1}{x} tends toward –∞ as x gets closer and closer to 0 from the left.

Formal Definitions:

1. lim⁡x→a−f(x)=L\displaystyle \lim_{x\to a^-} f(x) = L if and only if for each ε>0\varepsilon \gt 0 there exists a δ>0\delta \gt 0 such that
  a−δ<x<a⇒∣f(x)−L∣<ε.a - \delta \lt x \lt a \Rightarrow |f(x)-L| \lt \varepsilon\text{.}

2. lim⁡x→a−f(x)=∞\displaystyle \lim_{x\to a^-} f(x) = \infty if and only if for each NN there exists a δ>0\delta \gt 0 such that
  a−δ<x<a⇒f(x)>N.a - \delta \lt x \lt a \Rightarrow f(x) \gt N\text{.}

3. lim⁡x→a−f(x)=−∞\displaystyle \lim_{x\to a^-} f(x) = -\infty if and only if for each NN there exists a δ>0\delta \gt 0 such that
  a−δ<x<a⇒f(x)<N.a - \delta \lt x \lt a \Rightarrow f(x) \lt N\text{.}

See also

Limit from the right, infinity

Key Formula

lim⁡x→c−f(x)=L\lim_{x \to c^-} f(x) = L
Where:
  • xx = The input (domain) variable approaching the value c
  • cc = The value that x approaches, with x restricted to values less than c
  • f(x)f(x) = The function being evaluated
  • LL = The value that f(x) gets arbitrarily close to as x approaches c from the left
  • c−c^- = The superscript minus sign indicates the approach is from the left (from below)

Worked Example

Problem: Find the limit from the left of f(x) = 1/x as x approaches 0.
Step 1: Write the limit using left-hand limit notation.
lim⁡x→0−1x\lim_{x \to 0^-} \frac{1}{x}
Step 2: Choose values of x that are negative and getting closer to 0: x = −1, −0.1, −0.01, −0.001.
f(−1)=−1,f(−0.1)=−10,f(−0.01)=−100,f(−0.001)=−1000f(-1) = -1, \quad f(-0.1) = -10, \quad f(-0.01) = -100, \quad f(-0.001) = -1000
Step 3: As x approaches 0 from the left, the outputs become increasingly large in the negative direction.
1x→−∞ as x→0−\frac{1}{x} \to -\infty \text{ as } x \to 0^-
Step 4: State the result. The function does not approach a finite number; it decreases without bound.
lim⁡x→0−1x=−∞\lim_{x \to 0^-} \frac{1}{x} = -\infty
Answer: The limit from the left of 1/x as x approaches 0 is −∞-\infty.

Another Example

This example uses a piecewise function to show how the left-hand limit selects only the piece defined for x < c, and demonstrates a case where the left and right limits disagree.

Problem: Find the limit from the left of the piecewise function g(x) as x approaches 2, where g(x) = x + 3 for x < 2 and g(x) = 10 − x for x ≥ 2.
Step 1: Write the left-hand limit. Since we approach 2 from the left, we only use x < 2.
lim⁡x→2−g(x)\lim_{x \to 2^-} g(x)
Step 2: For x < 2, the rule is g(x) = x + 3. Substitute values approaching 2 from below: x = 1.9, 1.99, 1.999.
g(1.9)=4.9,g(1.99)=4.99,g(1.999)=4.999g(1.9) = 4.9, \quad g(1.99) = 4.99, \quad g(1.999) = 4.999
Step 3: The outputs approach 5. You can also substitute directly into x + 3.
lim⁡x→2−(x+3)=2+3=5\lim_{x \to 2^-}(x + 3) = 2 + 3 = 5
Step 4: Note that the limit from the right would use the other piece: g(x) = 10 − x gives 10 − 2 = 8. The left-hand and right-hand limits differ, so the two-sided limit does not exist at x = 2.
lim⁡x→2−g(x)=5≠8=lim⁡x→2+g(x)\lim_{x \to 2^-} g(x) = 5 \neq 8 = \lim_{x \to 2^+} g(x)
Answer: The limit from the left is 5. Because this does not equal the limit from the right (which is 8), the two-sided limit at x = 2 does not exist.

Frequently Asked Questions

What is the difference between a limit from the left and a limit from the right?
A limit from the left (x→c−x \to c^-) considers only x-values less than c, while a limit from the right (x→c+x \to c^+) considers only x-values greater than c. If both one-sided limits exist and are equal, the two-sided limit exists and equals that common value. If they differ, the two-sided limit does not exist.
When do you need to use a left-hand limit?
You use left-hand limits when analyzing piecewise functions at the boundary between pieces, when a function is only defined on one side of a point, or when determining whether a two-sided limit exists. They also arise when studying continuity: a function is continuous at c only if both one-sided limits equal f(c).
What does the minus sign in the superscript mean?
The superscript minus in c−c^- does not mean a negative number. It is notation indicating that x approaches c from values that are less than c — that is, from the left side on the number line. Similarly, c+c^+ means x approaches c from values greater than c.

Limit from the Left vs. Limit from the Right

Limit from the LeftLimit from the Right
Notationlim⁡x→c−f(x)\lim_{x \to c^-} f(x)lim⁡x→c+f(x)\lim_{x \to c^+} f(x)
Direction of approachx approaches c through values less than cx approaches c through values greater than c
Also calledLeft-hand limit, limit from belowRight-hand limit, limit from above
Piecewise functionsUses the piece defined for x < cUses the piece defined for x > c
Relationship to two-sided limitMust equal the right-hand limit for the two-sided limit to existMust equal the left-hand limit for the two-sided limit to exist

Why It Matters

Left-hand limits appear throughout calculus whenever you check continuity at a point or analyze piecewise-defined functions, which model real situations like tax brackets or shipping rates. They are essential for understanding the formal definition of a limit: the two-sided limit exists only when the left-hand and right-hand limits both exist and are equal. Mastering one-sided limits is also a prerequisite for topics like derivatives, improper integrals, and the behavior of functions near vertical asymptotes.

Common Mistakes

Mistake: Confusing the superscript minus with a negative sign and thinking x→0−x \to 0^- means x approaches −0 or a negative number.
Correction: The superscript −^- is directional notation meaning 'from below.' When you see x→0−x \to 0^-, it means x takes values like −0.1, −0.01, −0.001 — values slightly less than 0, not that x is heading toward some other number.
Mistake: Using the wrong piece of a piecewise function when computing the left-hand limit.
Correction: For lim⁡x→c−f(x)\lim_{x \to c^-} f(x), always use the rule that applies when x<cx < c, even if the function is defined differently at x=cx = c itself. The value f(c)f(c) does not matter for the limit.

Related Terms

  • One-Sided Limit — General category that includes left and right limits
  • Limit from the Right — The other one-sided limit, approaching from above
  • Domain — The set of x-values restricted in a left-hand limit
  • Variable — The input quantity x that approaches a value
  • Infinity — Possible result when a left-hand limit is unbounded
  • Continuous — Requires left and right limits to equal f(c)
  • Limit — Two-sided limit that exists when both one-sided limits agree