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Calculus Solver

Limit Calculator

Evaluate two-sided, one-sided and infinite limits. The calculator shows whether the limit was found by substitution, algebra or L'Hôpital's rule, and it checks every answer numerically from the side you asked about.

Your expression will be shown here as it will be read.
Direction

Type inf or -inf for limits at infinity (direction is ignored there).

Solved entirely in your browser. Nothing you type is sent to a server or stored, and no account is needed.

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What is a limit?

The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to when x gets close to a, without x ever having to equal a. That is why limits can exist at points where the function itself is undefined, such as (x² − 9)/(x − 3) at x = 3.

A one-sided limit only lets x approach from one direction: from the left (x → a⁻) or from the right (x → a⁺). The two-sided limit exists only when both one-sided limits exist and are equal. Limits at infinity describe the long-run behaviour of a function, such as horizontal asymptotes.

lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = L

Techniques for evaluating limits

The calculator applies these in order and stops at the first that settles the limit.

  • Direct substitution

    lim⁡x→af(x)=f(a)  if f is continuous at a\lim_{x\to a} f(x) = f(a)\ \text{ if } f \text{ is continuous at } a

    Always try this first. Polynomials, exponentials and trig functions are continuous wherever they are defined.

  • Factor and cancel

    lim⁡x→3x2−9x−3=lim⁡x→3(x+3)=6\lim_{x\to 3}\frac{x^{2}-9}{x-3} = \lim_{x\to 3}(x+3) = 6

    For a 0/0 rational function, cancel the common factor that causes the zero.

  • L'Hôpital's rule

    lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)for 00 or ∞∞\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}\quad\text{for } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}

    Differentiate the top and bottom separately. Not the quotient rule.

  • Degree comparison at infinity

    lim⁡x→∞anxn+⋯bmxm+⋯={0n<man/bmn=m±∞n>m\lim_{x\to\infty}\frac{a_n x^{n} + \cdots}{b_m x^{m} + \cdots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

    Only the leading terms matter for rational functions.

  • Other indeterminate forms

    0⋅∞→00,1∞, 00, ∞0→elim⁡(gln⁡f)0\cdot\infty \to \frac{0}{0},\qquad 1^{\infty},\ 0^{0},\ \infty^{0} \to e^{\lim (g \ln f)}

    Products are rewritten as quotients, and exponential forms are handled with logarithms.

How this calculator works

The calculator first probes the function numerically from the requested side or sides, evaluating it closer and closer to the point. If the left and right behaviour disagree, it reports that the two-sided limit does not exist and shows both one-sided values.

It then tries to evaluate the limit exactly: direct substitution, factor-and-cancel using a polynomial greatest common divisor, L'Hôpital's rule (repeated if needed), degree comparison for rational functions at infinity, and logarithmic rewriting for forms like 1^∞. Every exact answer must agree with the numerical probe before it is shown.

If no symbolic method completes, the numerical estimate is shown and clearly labelled. When the estimate matches a simple value such as π/2 or 1/2, that value is suggested as a strong estimate, not presented as a proof.

For the full details, including how answers are verified, see how the solver works.

Worked examples

Work through these by hand first, then enter them above to compare each step.

Example 1

Direct substitution

lim⁡x→2(x2+3x)\lim_{x\to 2}\left(x^{2} + 3x\right)

  1. Polynomials are continuous everywhere, so substitute x = 2.

    22+3⋅2=4+62^{2} + 3\cdot 2 = 4 + 6

Answer

1010

Example 2

Factor and cancel (0/0)

lim⁡x→3x2−9x−3\lim_{x\to 3}\frac{x^{2} - 9}{x - 3}

  1. Substitution gives 0/0, an indeterminate form. Factor the numerator.

    (x−3)(x+3)x−3=x+3(x≠3)\frac{(x-3)(x+3)}{x-3} = x + 3 \quad (x \neq 3)

  2. Now substitute x = 3.

Answer

66

Example 3

L'Hôpital's rule twice

lim⁡x→0ex−1−xx2\lim_{x\to 0}\frac{e^{x} - 1 - x}{x^{2}}

  1. At x = 0 this is 0/0. Differentiate numerator and denominator:

    lim⁡x→0ex−12x\lim_{x\to 0}\frac{e^{x} - 1}{2x}

  2. Still 0/0, so apply the rule again:

    lim⁡x→0ex2=12\lim_{x\to 0}\frac{e^{x}}{2} = \frac{1}{2}

Answer

12\frac{1}{2}

Example 4

Limit at infinity

lim⁡x→∞4x3−x2x3+5\lim_{x\to\infty}\frac{4x^{3} - x}{2x^{3} + 5}

  1. Divide every term by x³, the highest power in the denominator.

    lim⁡x→∞4−1/x22+5/x3=42\lim_{x\to\infty}\frac{4 - 1/x^{2}}{2 + 5/x^{3}} = \frac{4}{2}

Answer

22

Example 5

One-sided limits that disagree

lim⁡x→0∣x∣x\lim_{x\to 0}\frac{|x|}{x}

  1. For x > 0, |x|/x = 1. For x < 0, |x|/x = −1.

    lim⁡x→0+∣x∣x=1,lim⁡x→0−∣x∣x=−1\lim_{x\to 0^{+}}\frac{|x|}{x} = 1,\qquad \lim_{x\to 0^{-}}\frac{|x|}{x} = -1

  2. The one-sided limits differ, so the two-sided limit does not exist.

Answer

does not exist\text{does not exist}

Common mistakes

  • Reading 0/0 as 0 or as “undefined”

    0/0 is indeterminate: it tells you that more work is needed, not what the answer is. The limit can be any number, infinite, or nonexistent.

  • Using L'Hôpital's rule when the form is not 0/0 or ∞/∞

    For lim x→0 of (x + 1)/(x + 2), substitution gives 1/2. Applying L'Hôpital anyway would give 1/1 = 1, which is wrong.

  • Applying the quotient rule instead

    L'Hôpital's rule differentiates the numerator and denominator separately. It does not differentiate the fraction as a whole.

  • Ignoring direction

    At a vertical asymptote such as 1/x at 0, the left and right behaviour can differ. Always check both sides for a two-sided limit.

  • Treating ∞ as a number

    Writing lim = ∞ means the function grows without bound. The limit does not exist as a real number, and ∞ − ∞ is not 0.

Limitations

  • Piecewise-defined functions cannot be entered directly; evaluate each one-sided limit with the matching formula instead.
  • Limits that need series expansions or squeeze-theorem arguments may be answered numerically, with a clear label.
  • Strongly oscillating functions such as sin(1/x) near 0 are reported as undetermined rather than guessed.
  • Multivariable limits are not supported.

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Frequently asked questions

How do I enter a limit at infinity?

Type inf (or infinity, or ∞) in the “approaches” box. Use -inf for negative infinity. The direction setting is ignored at infinity.

What is the difference between a one-sided and a two-sided limit?

A one-sided limit only looks at values of x on one side of the point. The two-sided limit exists only when both one-sided limits exist and agree.

When does a limit not exist?

When the one-sided limits differ (a jump), when the function oscillates without settling (sin(1/x) near 0), or when it grows without bound. In that last case we often write ±∞ to describe how it fails to exist.

Why does the calculator sometimes say “numerical estimate”?

If none of the symbolic methods settles the limit, the calculator evaluates the function very close to the point instead. That is reliable for well-behaved functions, but it is not a proof, so the answer is labelled accordingly.

Is L'Hôpital's rule always the best method?

No. It only applies to 0/0 and ∞/∞ forms, and factoring or simplifying is often faster and clearer. The calculator tries substitution and factoring first.

Next step

Pick one of the worked examples, solve it on paper, then enter it in the calculator and compare your steps with each rule shown. Spotting where your work differs is the fastest way to improve.

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