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One-Sided Limit Calculator

Evaluate one-sided limits from the left or right. Identify jump discontinuities and points where only one-sided limits exist.

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Documentation

One-Sided Limits

A one-sided limit considers the behavior of $f(x)$ as $x$ approaches a value from only one direction.

Left-Hand Limit

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

Considers only values of $x$ less than $a$ (approaching from the left).

Right-Hand Limit

\lim_{x \to a^{+}} f(x) = L^{+}

Considers only values of $x$ greater than $a$ (approaching from the right).

Key theorem: The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal: $L^{-} = L^{+}$ .

Jump Discontinuities

When $L^{-} \neq L^{+}$ , the function has a jump discontinuity at $x = a$ . The two-sided limit does not exist, but both one-sided limits are well-defined. Classic examples:

Piecewise functions (e.g., step functions, absolute value at the origin)
The floor function $\lfloor x \rfloor$ at every integer
The sign function $\text{sgn}(x)$ at $x = 0$

Worked Examples

Example: Absolute Value

For $f(x) = |x|/x$ :

\lim_{x \to 0^{-}} \frac{|x|}{x} = -1, \qquad \lim_{x \to 0^{+}} \frac{|x|}{x} = +1

The one-sided limits differ, so the two-sided limit does not exist.

Example: Piecewise Function

For $f(x) = \begin{cases} x^2 & x < 1 \\ 2x - 1 & x \geq 1 \end{cases}$ :

\lim_{x \to 1^{-}} f(x) = 1, \qquad \lim_{x \to 1^{+}} f(x) = 1

Both one-sided limits equal 1, so the two-sided limit exists and equals 1. Since $f(1) = 1$ as well, the function is continuous at $x = 1$ .

Frequently Asked Questions

What is a one-sided limit?

A one-sided limit examines the behavior of $f(x)$ as $x$ approaches a value from only one direction. The left-hand limit ( $x \to a^-$ ) is written with a superscript minus, and the right-hand limit ( $x \to a^+$ ) with a superscript plus.

When does a two-sided limit exist?

The two-sided limit exists if and only if both one-sided limits exist and are equal. If the left-hand limit and right-hand limit give different values, the two-sided limit does not exist.

What is a jump discontinuity?

A jump discontinuity occurs at a point where the left-hand limit and right-hand limit both exist but are not equal. The function “jumps” from one value to another. Examples include step functions, the floor function at integers, and the sign function at zero.

How do I evaluate one-sided limits for piecewise functions?

Use the piece of the function that applies to the direction you are approaching from. For the left-hand limit, use the formula valid for values less than the point. For the right-hand limit, use the formula valid for values greater than the point.

Can a function be continuous if one-sided limits differ?

No. A function is continuous at a point only if the two-sided limit exists (meaning both one-sided limits are equal) and equals the function value at that point. If one-sided limits differ, the function has a jump discontinuity and is not continuous there.

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