One-Sided Limit Calculator

The one-sided limit calculator finds the left-hand limit \( \lim_{x\to c^-} f(x) \), the right-hand limit \( \lim_{x\to c^+} f(x) \), or the ordinary two-sided limit, for any function you type. It also handles limits as \( x \to \infty \) and \( x \to -\infty \), which are one-sided by nature.

Use it when a graph has a jump, a vertical asymptote or a domain edge, and you need to know exactly what happens on each side. It is free and runs in your browser with no sign-up.

How to use the one-sided limit calculator

1. Type \( f(x) \). Use ^ for powers and sqrt(), abs(), sin(), ln(), e^(x) and pi as needed. Implicit multiplication like 2x or x e^x works.
2. Enter the point \( x \) approaches. Use inf or -inf for limits at infinity.
3. Pick the direction: From the left approaches from values smaller than \( c \), From the right from values larger than \( c \), and Two-sided requires both to agree. The direction is ignored for inf and -inf.
4. Press Evaluate limit. The output gives the limit, the method used (direct substitution, L’Hôpital’s rule or a numerical estimate) and, when a two-sided limit fails, the left and right values side by side.

How one-sided limits work

The left-hand limit only looks at \( x < c \), and the right-hand limit only at \( x > c \):

$$\lim_{x\to c^-} f(x) = L_1, \qquad \lim_{x\to c^+} f(x) = L_2$$

The two-sided limit exists exactly when both one-sided limits exist and are equal:

$$\lim_{x\to c} f(x) = L \iff L_1 = L_2 = L$$

When substitution and L’Hôpital’s rule don’t settle the question, the calculator samples \( f \) at \( c - 0.1, c - 0.01, \dots \) down to a distance of \( 10^{-9} \) for the left side (and the mirror points for the right side). For \( x \to \pm\infty \) it samples \( x = \pm10, \pm100, \dots \) up to \( \pm10^8 \). It then reports the value the outputs settle on, or \( \pm\infty \) if they grow without bound. This is a careful numerical estimate, so back it up with algebra when you need a proof.

Worked example

Example 1: a vertical asymptote. Find both one-sided limits of \( f(x) = \frac{1}{x} \) at 0.

For small positive \( x \) the fraction is large and positive: \( f(0.001) = 1000 \). For small negative \( x \) it is large and negative: \( f(-0.001) = -1000 \). So

$$\lim_{x\to0^+}\frac1x = \infty, \qquad \lim_{x\to0^-}\frac1x = -\infty.$$

The sides disagree, so the two-sided limit does not exist. Type 1/x with \( x \to 0 \) into the calculator above and switch between the three directions to see all three answers.

Example 2: sides that behave differently. Consider \( g(x) = e^{1/x} \) at 0.

From the right, \( \frac1x \to \infty \), so \( e^{1/x} \to \infty \). From the left, \( \frac1x \to -\infty \), so \( e^{1/x} \to 0 \). One side has a finite limit and the other blows up:

$$\lim_{x\to0^-} e^{1/x} = 0, \qquad \lim_{x\to0^+} e^{1/x} = \infty.$$

Example 3: a limit at negative infinity. Find \( \lim_{x\to-\infty} x e^x \). The factor \( x \) grows in size, but \( e^x \) shrinks much faster, so the product goes to 0. Enter x e^x with -inf to check. At \( +\infty \) both factors grow and the limit is \( \infty \).

Where one-sided limits show up

• Rational functions near a zero of the denominator. For \( \frac{2x+1}{x-3} \) at 3, the numerator tends to 7 while the denominator is a small positive number on the right and a small negative number on the left. So the limit is \( \infty \) from the right and \( -\infty \) from the left. A sign chart predicts this before you compute anything.
• Absolute value and piecewise formulas. \( \frac{|x|}{x} \) equals \( -1 \) for \( x < 0 \) and \( 1 \) for \( x > 0 \), so its left and right limits at 0 are \( -1 \) and \( 1 \).
• Domain edges. Functions like \( \sqrt{x} \) or \( \ln x \) only exist on one side of 0, so only the right-hand limit makes sense there.
• Continuity questions. A function is continuous at \( c \) when both one-sided limits equal \( f(c) \). Checking each side separately is how you classify jump and infinite discontinuities.

For end behavior and horizontal asymptotes, the guide to limits at infinity goes through the degree rules. If a one-sided limit gives \( \tfrac00 \), L’Hôpital’s rule still applies on that side, and the L’Hôpital’s rule calculator shows each round. For ordinary two-sided limits with numbered steps, use the limit calculator.

Related guides

• Limits at Infinity: Rules, Asymptotes and Examples
• L’Hôpital’s Rule: When and How to Use It
• Squeeze Theorem: Statement and Examples
• Calculus Concepts Explained

Further reading

• One-Sided Limits (Paul’s Online Math Notes) — left- and right-hand limits and when the two-sided limit exists.
• Infinite Limits (Paul’s Online Math Notes) — one-sided limits that blow up, and vertical asymptotes.

FAQ

What is the difference between a left-hand and a right-hand limit?

The left-hand limit uses only inputs smaller than \( c \), written \( x \to c^- \). The right-hand limit uses only inputs larger than \( c \), written \( x \to c^+ \).

When does a two-sided limit not exist?

It fails when the one-sided limits are different, when either one doesn’t exist, or when the sides go to infinities of opposite sign, as with \( \frac1x \) at 0.

Can a one-sided limit be infinite?

Yes. \( \lim_{x\to0^+}\frac1x = \infty \) is a one-sided limit that tells you the graph has a vertical asymptote at \( x = 0 \), approached upward from the right.

Does the calculator find limits at infinity?

Yes. Enter inf or -inf as the point. The direction setting is ignored there, because \( x \) can only approach infinity from one side.