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Home›Function Tools›One-Sided Limit Calculator

One-Sided Limit Calculator

Compute left-hand and right-hand limits of rational and trig functions

Select Function Type

Rational (ax+b)/(cx+d) Piecewise at boundary

Evaluate as x approaches p from left and right

lim x->(x+)/(x+

One-Sided Limits

Left limit: lim x->a- f(x) (approach from below)

Right limit: lim x->a+ f(x) (approach from above)

Two-sided limit exists iff both sides equal

If they differ: jump discontinuity

One-sided limits examine function behavior approaching a point from a single direction. The left limit uses x values slightly less than a. The right limit uses x values slightly greater. Comparing them reveals continuity and discontinuity type.

⚠ The two-sided limit exists only when left and right limits are equal and finite. If they differ or are infinite, the limit does not exist.

What Are One-Sided Limits?

One-sided limits describe behavior approaching a point from one direction. They are essential for analyzing piecewise functions, vertical asymptotes, and discontinuities. The left and right limits may differ at jumps or be infinite at asymptotes.

Left Limit

Approach from smaller values. x -> a- means x = a - epsilon where epsilon is a small positive number.

Right Limit

Approach from larger values. x -> a+ means x = a + epsilon. The function may behave differently on each side.

Continuity Check

f continuous at a if lim x->a- = lim x->a+ = f(a). If one-sided limits differ, the function has a jump.

Infinite Limits

If f(x) goes to +/-infinity when approaching from one or both sides, the limit is infinite (vertical asymptote).

Teaching Example: lim x->2 (x-4)/(x-2). Left: values just below 2 give ((-epsilon)-4)/(-epsilon) -> +inf. Right: values just above 2 give ((epsilon)-4)/(epsilon) -> -inf. Left != Right, so two-sided limit DNE.

Applications

Continuity Asymptotes Piecewise Calculus Derivatives

Frequently Asked Questions

What is one-sided limit?▼

Lim x->a-: approach from below. Lim x->a+: approach from above. One direction only.

When does limit exist?▼

When left = right limit. Both must be finite and equal. Otherwise the two-sided limit DNE.

Jump discontinuity?▼

Occurs when left and right limits are finite but not equal. The function jumps from one value to another at a.

Infinite one-sided limit?▼

When f(x) approaches +/-inf from one or both sides. This indicates a vertical asymptote.

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