Limit Calculator: Limit of f(x) as x Approaches a Number or Gets Very Large, from the Left, Right or Both Sides

Numerical limits, including one-sided and at large x.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Type f(x) and the value a.

Choose both sides, one side or very large x.

The calculator shows the values near a and the estimated limit.

Formulas used

  • \lim_{x\to a}f(x) = L
  • \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x)
  • \lim_{x\to\infty}f(x)

Worked example

With the default numbers in the calculator above:

  1. Value just left of a (at a - 0.00001): 1
  2. Value just right of a (at a + 0.00001): 1
  3. Limit (estimate): 1

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: f(x) (1-cos(x))/x^2.

  1. Value just left of a (at a - 0.00001): 0.5
  2. Value just right of a (at a + 0.00001): 0.5
  3. Limit (estimate): 0.5

Common questions

When does a limit not exist?

When the left and right values are different, when the values grow without limit, or when they keep oscillating.

Can a limit exist where the function is not defined?

Yes. The limit only looks at values near the point, not at the point itself.

Is this exact?

It is a numerical estimate from values very close to the point. For exact work use algebra or L'Hopital's rule.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any calculus text (limits of functions).

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