This learning article is currently available in English. It is not published as a translated page.

One-sided limits

When Left and Right Limits Differ, the Limit Does Not Exist

A two-sided limit makes two claims at once: values approaching from the left and from the right must settle at the same target. Finding only one side is not enough, and a function value at the point cannot repair a directional mismatch.

Direct answer

A two-sided limit exists only when the left-hand and right-hand limits approach the same value.

Check both directions separately whenever the expression changes behavior across the approach point.

Worked comparison: 1/x as x approaches 0

For negative x close to zero, 1/x is negative with increasing magnitude, so the left-hand limit is −∞. For positive x close to zero, 1/x is positive with increasing magnitude, so the right-hand limit is +∞.

Because the two directions do not agree, the two-sided limit does not exist. Writing “∞” for the two-sided result would hide the opposite signs and is therefore incorrect.

The same test handles finite jumps. If the left side approaches 2 and the right side approaches 5, neither the function value nor an average of 2 and 5 becomes the limit. Agreement is required.

A removable hole is different. If both sides approach the same number, the limit may exist even when the function is undefined there or has a different assigned value.

1
lim x→0⁻ 1/x = −∞
2
lim x→0⁺ 1/x = +∞
3
Different directions → the two-sided limit does not exist

Worked examples

lim x→0⁻ 1/x
−∞

Approaching through negative inputs keeps the reciprocal negative.

lim x→0⁺ 1/x
∞\infty

Approaching through positive inputs keeps the reciprocal positive.

lim x→1 (x²−1)/(x−1)
22

After cancellation, both sides approach the same value even though the original expression has a hole.

Common mistakes

  • • Checking only the side that is easier to evaluate.
  • • Using the function value at the point as the limit without examining nearby values.
  • • Calling opposite infinite directions one common infinite limit.
  • • Assuming every undefined function value makes the limit nonexistent.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

极限:1x\frac{1}{x}极限:1x\frac{1}{x}极限:1x\frac{1}{x}
Open the calculator without a preset: 极限

References

Frequently asked questions

Can a two-sided limit exist when the function is undefined?+

Yes. The limit depends on nearby values. If both one-sided limits agree, a missing function value does not prevent the limit from existing.

What if one side is finite and the other is infinite?+

The directions disagree, so the two-sided limit does not exist.

Does a jump discontinuity always destroy the two-sided limit?+

At the jump point, yes: the left and right sides approach different finite values. Away from that point, other limits may still exist.

Conclusion

Compute both directions before reporting a two-sided limit. Matching values establish the limit; any directional disagreement means the two-sided limit does not exist at that point.

Compare usage plans