Limit Calculator
Compute numerical limits of real-valued functions: left-hand, right-hand and two-sided limits, plus limits at infinity. Supports sin, cos, ln, sqrt, powers and more, with examples and explanations.
Full original guide (expanded)
Limit Calculator
Numerical left, right & two-sided limitsExplore 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
Use x as the variable. Allowed: +, -, *, /, ^, parentheses, sin, cos, tan, ln, log, exp, sqrt, abs, floor, ceil, pi, e.
For limits at ±∞, the point a is ignored.
Ignored when taking limits at ±∞.
More points = smoother pattern, but slower.
Display precision for numerical values.
2. Numerical result and classification
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.
| x | f(x) |
|---|
| 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.
Formula (LaTeX) + variables + units
\lim_{x \to a} f(x) = L
\lim_{x \to a^-} f(x)
\lim_{x \to a^+} f(x)
','\
\[ \lim_{x \to a} f(x) = L \] 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.
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).
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Last code update: 2026-01-19
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