Softonic

Limits Calculator

Step 1 / 3 Limit value
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The limit of a function f(x) as x approaches a point a describes the value f(x) settles toward, whether or not f(a) itself exists. This calculator evaluates limits numerically and symbolically for polynomials, rational functions, trigonometric functions, exponentials and logs, including points at ±∞, and tells you when the two-sided limit does not exist because left and right values disagree.

How to compute a limit

  1. 1

    Pick a function family

    Polynomial, rational, trig (sin/cos/tan), exponential, log or a combination.

  2. 2

    Enter the limit point a

    A real number, or use ∞ / -∞ for limits at the extremes.

  3. 3

    Choose a direction

    Two-sided (both), left (a⁻) or right (a⁺).

  4. 4

    Read the numeric result

    If left and right disagree, the two-sided limit does not exist.

When limits behave and when they don't

Most first-year calculus problems rely on three techniques: direct substitution, algebraic simplification, and L’Hôpital’s rule. The calculator applies them in that order and falls back to a numerical estimate when symbolic manipulation fails.

Classic results worth memorising

Limit Value
lim x→0 sin(x)/x 1
lim x→0 (1 − cos x)/x 0
lim x→0 (1 − cos x)/x² 1/2
lim x→0 (eˣ − 1)/x 1
lim x→∞ (1 + 1/x)ˣ e
lim x→0⁺ x ln(x) 0
lim x→∞ ln(x)/x 0
lim x→∞ (polynomial)/eˣ 0

When the two-sided limit fails

Three patterns cause “does not exist”:

  • Jump discontinuity. f(x) = sign(x) at x = 0: left → -1, right → +1.
  • Vertical asymptote with opposite signs. f(x) = 1/x at x = 0: left → -∞, right → +∞.
  • Oscillation. f(x) = sin(1/x) at x = 0: values oscillate between -1 and 1 arbitrarily close to the point.

One-sided limits may exist where two-sided limits do not. The calculator reports each side separately so you can see the divergence.

L’Hôpital’s rule: use with care

When a substitution gives 0/0 or ∞/∞, differentiating numerator and denominator separately often resolves it. But the rule requires:

  • Both limits (of numerator and denominator) must genuinely be 0 or both ∞.
  • The derivative ratio must itself have a limit.
  • The functions must be differentiable near the point.

Misapplying L’Hôpital to a non-indeterminate form gives wrong answers. Always confirm the form first.

Numerical vs symbolic

The calculator uses exact arithmetic for simple algebraic forms (polynomials, rationals). For transcendental expressions it evaluates the function at a sequence of points approaching a and reports the converged value to 8 decimal places. If the sequence fails to converge, the result is flagged as indeterminate.

Frequently Asked Questions

The value f(x) approaches when x comes from only one side of a. Left-sided (x → a⁻) uses values below a; right-sided (x → a⁺) uses values above. The two-sided limit exists only when both match.

Yes, enter ∞ or -∞ as the limit point. For rational functions it compares degrees; for exponentials and logs it uses the standard asymptotic rankings.

The function does not settle to a single finite value as x approaches the point. This can happen because values tend to infinity, because left and right disagree, or because values oscillate without converging.

No. Expressions you enter are parsed and evaluated in your browser and nothing is sent anywhere.

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