Can a function have a two-sided limit if the one-sided limits are different?

No. By definition, \lim_{x \to a} f(x) exists if and only if \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x). If the one-sided limits are unequal, there is no two-sided limit.

What does the - in x \to a^- mean?

It means x approaches a from the left—using only values of x that are less than a. Similarly, x \to a^+ means approaching from the right, using values greater than a. The superscripts indicate the direction of approach.

Can the limit exist if the function is undefined at the point?

Yes. If both one-sided limits exist and are equal, the two-sided limit exists even if f(a) is undefined or does not equal the limit value. This situation is called a removable discontinuity.

How do I compute a one-sided limit for a piecewise function?

Identify which piece of the function applies as you approach from that direction: for x \to a^-, use the formula defined for x a. Then substitute to evaluate the limit.

One-Sided Limits

One-sided limits let you examine how a function behaves as you approach a point from just the left or right, determining whether a two-sided limit exists.

\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \iff \lim_{x \to a} f(x) \text{ exists}

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What each symbol means

What One-Sided Limits takes

Symbol	Meaning
$f$	The function being analyzed; it takes input $x$ and produces an output value. Misreading $f$ as a number rather than a rule breaks limit notation.
$x$	The input variable approaching $a$ ; the superscript $-$ in $a^-$ means from the left, and $+$ means from the right. Reversing the superscript reverses which limit you compute.
$a$	The point toward which $x$ approaches; a specific real number. Confusing $a$ with $f(a)$ is a common error—the limit can exist even if $f(a)$ is undefined.

When to use it

Use one-sided limits when testing whether a regular limit exists or when a function behaves differently approaching from different directions.

Level

Usually taught in: Calculus I

Worked examples

1. Finding a two-sided limit from equal one-sided limits

Problem

Determine if

\lim_{x \to 3} f(x)

exists for the piecewise function where

f(x) = 2x

when

x < 3

and

f(x) = 2x

when

x > 3

$\lim_{x \to 3^-} f(x) = 2 \cdot 3 = 6$
When approaching 3 from the left where $f(x) = 2x$ , substitute $x = 3$ to get $2 \cdot 3 = 6$ .
$\lim_{x \to 3^+} f(x) = 2 \cdot 3 = 6$
When approaching from the right where $f(x) = 2x$ , substitute $x = 3$ to get $2 \cdot 3 = 6$ .
$6 = 6 \text{ so } \lim_{x \to 3} f(x) = 6$
Since both one-sided limits equal 6, the two-sided limit exists and equals 6.

Answer: $\lim_{x \to 3} f(x) = 6$

This problem shows the key principle: when the left and right one-sided limits are equal, the two-sided limit exists. Both pieces of the function approach the same value, so we confidently conclude the limit is 6.

2. Detecting a jump discontinuity with unequal one-sided limits

Problem

Determine if

\lim_{x \to 2} f(x)

exists for the piecewise function where

f(x) = x + 1

when

x < 2

and

f(x) = 2x + 1

when

x > 2

$\lim_{x \to 2^-} f(x) = 2 + 1 = 3$
Using the left piece $f(x) = x + 1$ for $x < 2$ , substitute $x = 2$ to get $2 + 1 = 3$ .
$\lim_{x \to 2^+} f(x) = 2 \cdot 2 + 1 = 5$
Using the right piece $f(x) = 2x + 1$ for $x > 2$ , substitute $x = 2$ to get $2 \cdot 2 + 1 = 5$ .
$3 \neq 5 \text{ so } \lim_{x \to 2} f(x) \text{ does not exist}$
Because the one-sided limits are unequal, the two-sided limit does not exist at $x = 2$ .

Answer: $\lim_{x \to 2} f(x) \text{ does not exist}$

This example demonstrates that when a function approaches different values from the left and right, a jump discontinuity occurs and the two-sided limit fails to exist. One-sided limits are essential for detecting such behavior in piecewise functions.

3. Analyzing a real-world discontinuity in a phone plan cost model

Problem

A phone plan charges $20 monthly plus $5 per gigabyte. If data usage exceeds 5 gigabytes, a $10 overage fee is added. Find if

\lim_{d \to 5} C(d)

exists, where

C(d)

is the total monthly cost for

d

gigabytes used.

$\lim_{d \to 5^-} C(d) = 20 + 5 \cdot 5 = 45$
For data usage just below 5 GB, the cost is $20 + 5 \cdot 5 = 45$ dollars.
$\lim_{d \to 5^+} C(d) = 30 + 5 \cdot 5 = 55$
For data usage just above 5 GB, the overage fee applies, so the cost is $30 + 5 \cdot 5 = 55$ dollars.
$45 \neq 55 \text{ so } \lim_{d \to 5} C(d) \text{ does not exist}$
Since the one-sided limits differ by $10, the cost has a jump discontinuity at 5 GB and the limit does not exist.

Answer: $\lim_{d \to 5} C(d) \text{ does not exist}$

This real-world example shows how one-sided limits reveal discontinuities in practical functions. The cost function jumps when data crosses 5 GB due to the overage rule. Even though both one-sided limits are finite, they are unequal, so the two-sided limit is undefined.

Common mistakes

Where One-Sided Limits usually goes wrong

The mistake	Why it is wrong	The fix
If $f(a)$ is defined, then $\lim_{x \to a} f(x)$ exists.	The existence of a two-sided limit depends only on whether the one-sided limits are equal, not on whether $f(a)$ is defined.	Check that $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ . If they are equal, the two-sided limit exists regardless of the value of $f(a)$ .
If the left and right one-sided limits are not equal, the function must have a jump discontinuity at $a$ .	Unequal one-sided limits can occur at various types of discontinuities, not just jump discontinuities (for example, at infinite discontinuities or corners with vertical behavior).	Unequal one-sided limits mean the two-sided limit does not exist; determine the type of discontinuity separately by examining whether the one-sided limits are finite or infinite.
You only need to check one one-sided limit to determine if the two-sided limit exists.	By definition, the two-sided limit exists only when both one-sided limits exist and are equal. Checking just one side is not sufficient.	Always compute both $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ and verify that they are equal before concluding the two-sided limit exists.

Tips and when to use something else

To find one-sided limits from a graph, look only at the curve approaching from one direction—imagine covering the other side with your hand.
If both one-sided limits are finite and equal, they equal the two-sided limit, even if the function is undefined or discontinuous at that point (see Removable Discontinuity for this case).
One-sided limits are powerful for detecting jump discontinuities and studying functions with corner points; for smooth functions, direct substitution usually works (see Limit Laws for a faster approach).
When a piecewise function changes rules at a point, always use the appropriate piece for each one-sided approach: apply the left piece for $x \to a^-$ and the right piece for $x \to a^+$ .

Related formulas

Definition of a LimitDefinition of a Limit describes what value a function approaches as its input approaches a specific point—essential for all of calculus.ContinuityContinuity tells you when a function has no jumps or breaks, so you can safely evaluate limits and apply theorems about function behavior.Removable DiscontinuityA removable discontinuity is a hole in a graph where the limit exists but the function value differs; it can be fixed by redefining that point.Infinite LimitsInfinite Limits describe when a function grows unboundedly as the input approaches a specific value, letting you identify and analyze vertical asymptotes.Limit LawsThe Sum Law shows that a limit of a sum equals the sum of the limits, provided both limits exist and are finite numbers.

Frequently asked questions

Can a function have a two-sided limit if the one-sided limits are different?: No. By definition, $\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ . If the one-sided limits are unequal, there is no two-sided limit.
What does the $-$ in $x \to a^-$ mean?: It means $x$ approaches $a$ from the left—using only values of $x$ that are less than $a$ . Similarly, $x \to a^+$ means approaching from the right, using values greater than $a$ . The superscripts indicate the direction of approach.
Can the limit exist if the function is undefined at the point?: Yes. If both one-sided limits exist and are equal, the two-sided limit exists even if $f(a)$ is undefined or does not equal the limit value. This situation is called a removable discontinuity.
How do I compute a one-sided limit for a piecewise function?: Identify which piece of the function applies as you approach from that direction: for $x \to a^-$ , use the formula defined for $x < a$ ; for $x \to a^+$ , use the formula for $x > a$ . Then substitute to evaluate the limit.

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Reviewed 2026-09-18