What is a limit in calculus?

The limit of a function f(x) as x approaches a value a describes the number that f(x) gets arbitrarily close to when x is taken sufficiently close to a. Limits formalise the idea of approaching a value without necessarily reaching it and are the foundation of derivatives and continuity.

What is the difference between left-hand and right-hand limits?

The left-hand limit looks at values of x that approach a from below, written as x → a−, while the right-hand limit looks at values approaching a from above, written as x → a+. A two-sided limit exists and equals L only if both the left-hand and right-hand limits exist and are equal to L.

Can this calculator compute exact symbolic limits?

No. This tool uses numerical sampling to approximate limits and detect patterns such as finite limits, divergence to infinity or oscillation. It is very useful for intuition, checking examples and teaching, but it does not replace formal symbolic methods in a computer algebra system or by hand.

How should I write functions for the limit calculator?

Write your function using x as the variable and standard operators +, −, *, / and ^ for powers, together with mathematical functions such as sin, cos, tan, ln, log, sqrt, exp, abs, floor and ceil. You can also use constants pi and e. For example: sin(x)/x, (x^2-1)/(x-1), abs(x)/x or (1+1/x)^x.

Is this calculator suitable for exam or professional use?

The calculator is designed as an educational and exploratory tool. It can help you build intuition and perform quick numerical checks, but for exam solutions, formal proofs or high-stakes engineering work you should always perform an analytical calculation or use a dedicated computer algebra system.

Limit Calculator

Numerical left, right & two-sided limits

Explore the limit of a real-valued function as x approaches a point or infinity. This tool approximates left-hand, right-hand and two-sided limits, detects divergence and oscillation, and displays sample values close to the point.

Good for: calculus homework, teaching intuition, checking discontinuities, removable singularities, asymptotic behaviour and left/right continuity in real-world formulas.

1. Define the limit

Function f(x)

Use x as the variable. Allowed: +, -, *, /, ^, parentheses, sin, cos, tan, ln, log, exp, sqrt, abs, floor, ceil, pi, e.

Limit type

Two-sided: x → a Left-hand: x → a⁻ Right-hand: x → a⁺ x → +∞ x → −∞

For limits at ±∞, the point a is ignored.

Point a (x → a)

Ignored when taking limits at ±∞.

Sample points per side

More points = smoother pattern, but slower.

Decimals

Display precision for numerical values.

Quick examples:

2. Numerical result and classification

Left-hand limit (x → a⁻)

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Right-hand limit (x → a⁺)

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Two-sided limit (candidate)

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Behaviour classification

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Function value at a (if defined)

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Approx. rate of approach

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Warnings

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3. Sample values near the point

The table below shows how f(x) behaves as x approaches the point from the left and from the right. Use it to spot removable discontinuities, jumps and oscillations.

Left side (x < a) Smaller |x−a| → closer to the point

x	f(x)

Right side (x > a)

x	f(x)

Conceptual definition of a limit

Intuitively, the limit of f(x) as x approaches a value a is the number L (if it exists) that f(x) gets arbitrarily close to whenever x is taken sufficiently close to a, but not necessarily equal to a. Formally, we write:

\[ \lim_{x \to a} f(x) = L \]

if for every tolerance \(\varepsilon > 0\) there exists a distance \(\delta > 0\) such that whenever 0 < |x − a| < δ we have |f(x) − L| < ε.

The numerical approach used here samples points closer and closer to a to approximate how f(x) behaves, which is excellent for building intuition but does not replace the formal ε–δ definition.

Left-hand, right-hand and two-sided limits

Sometimes the behaviour of a function differs depending on whether x approaches a from below or from above. We distinguish:

Left-hand limit: \[ \lim_{x \to a^-} f(x) \]
Right-hand limit: \[ \lim_{x \to a^+} f(x) \]
The two-sided limit \(\lim_{x \to a} f(x)\) exists and equals L if and only if both one-sided limits exist and are equal to L.

The calculator always computes both directions (where applicable) and shows whether they appear to agree numerically.

Limits at infinity and asymptotic behaviour

Limits as x tends to +∞ or −∞ capture the asymptotic behaviour of a function, including horizontal asymptotes and growth rates. Numerically, we approximate them by evaluating f(x) at larger and larger |x| and checking whether the values stabilise, diverge or oscillate.

Examples

\(\displaystyle \lim_{x\to 1} \frac{x^2 - 1}{x - 1} = 2\) (removable discontinuity).
\(\displaystyle \lim_{x\to 0} \frac{\sin x}{x} = 1\) (classic trigonometric limit).
\(\displaystyle \lim_{x\to 0} \frac{|x|}{x}\) does not exist, because left and right limits are −1 and +1 respectively (jump discontinuity).
\(\displaystyle \lim_{x\to +\infty} \left(1+\frac{1}{x}\right)^x = e\) (definition of e).

FAQ: using this limit calculator correctly

How reliable is the numerical classification?

The algorithm looks at sample points close to the approach point (or large |x| for infinity limits) and checks whether values are converging, diverging or oscillating. For well-behaved functions this is usually accurate, but exotic or highly oscillatory functions may require more points or a dedicated analytical treatment.

What if I get NaN or “undefined” values?

This happens when the expression has a division by zero, a square root of a negative number, a logarithm of a non-positive number, or other domain issues. The calculator automatically discards undefined points, but if too many samples are invalid it will warn you and classification may not be meaningful.

Can I use degrees instead of radians?

Trigonometric functions (sin, cos, tan, etc.) follow the JavaScript convention and expect radians. If you need degrees, convert them manually: for example, sin(x*pi/180) to interpret x as degrees.

Does this replace a symbolic CAS for serious work?

No. This calculator is intentionally numerical and designed for rapid exploration, didactics and sanity checks. For rigorous derivations and exact expressions you should still use a computer algebra system or do a detailed hand calculation.