Limit calculator

Graph

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Formula

How it works

A limit exists at a point if the function approaches the same value whether x approaches from below or from above. This is checked numerically here by evaluating f very close to a on each side; where f itself is undefined at a (as with the reciprocal family at x = 0), the two one-sided values are shown instead.

Worked example

The same calculation with the default values:

• Function family Quadratic: f(x) = ax² + bx + c
• Coefficient a 1
• Coefficient b 0
• Constant c 0
• Approach x = 0

1. x → -0.0001⁻: f(x) ≈ 0
2. x → 0.0001⁺: f(x) ≈ 0
3. f(0) = 0

Result: Approaching from the left: 0.00000001 · Approaching from the right: 0.00000001 · Limit: 0

Common mistakes

• Assuming a limit always equals the function's value at that exact point - a limit describes the value the function approaches, which can differ from (or be defined where the function itself is not, such as a removable discontinuity).
• Using a step size too large when numerically approximating a limit - this can make a function that actually diverges look like it converges, or vice versa, if the sampled points aren't close enough to the target.
• Confusing a one-sided limit with the full (two-sided) limit - a function can approach different values from the left and right of a point, in which case the full limit does not exist even though each one-sided limit does.