Limit from the Left — Definition, Formula & Examples

Q: What is the difference between a limit from the left and a limit from the right?

A limit from the left ($x \to c^-$) considers only x-values less than c, while a limit from the right ($x \to c^+$) 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.

Q: 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).

Q: What does the minus sign in the superscript mean?

The superscript minus in $c^-$ 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, $c^+$ means x approaches c from values greater than c.

Limit from the Left
Limit from Below

A one-sided limit which, in the example , 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 or .

For example, since tends toward –∞ as x gets closer and closer to 0 from the left.

See also

Limit from the right, infinity

Key Formula

\lim_{x \to c^-} f(x) = L

Where:

$x$ = The input (domain) variable approaching the value c
$c$ = The value that x approaches, with x restricted to values less than c
$f(x)$ = The function being evaluated
$L$ = The value that f(x) gets arbitrarily close to as x approaches c from the left
$c^-$ = 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 \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, \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.

\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 \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 \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, \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 \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 \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 \to c^-

) considers only x-values less than c, while a limit from the right (

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^-

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^+

means x approaches c from values greater than c.

Limit from the Left vs. Limit from the Right

	Limit from the Left	Limit from the Right
Notation	$\lim_{x \to c^-} f(x)$	$\lim_{x \to c^+} f(x)$
Direction of approach	x approaches c through values less than c	x approaches c through values greater than c
Also called	Left-hand limit, limit from below	Right-hand limit, limit from above
Piecewise functions	Uses the piece defined for x < c	Uses the piece defined for x > c
Relationship to two-sided limit	Must equal the right-hand limit for the two-sided limit to exist	Must 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 \to 0^-

means x approaches −0 or a negative number.

Correction: The superscript

^-

is directional notation meaning 'from below.' When you see

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 \to c^-} f(x)

, always use the rule that applies when

x < c

, even if the function is defined differently at

x = c

itself. The value

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