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Limit Definition Calculator

Limit-definition demo. Enter f(x) and a to watch f(a±h) approach L as h shrinks, visualizing the epsilon-delta process—for beginners.

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lim(x→a) f(x) = L
Supports + - * / ^ ( ) and sin cos tan sqrt exp log pi e
x approaches this point (may be 0)

📖 Tutorial | Definition of a Limit

1. Definition
Let f be defined near a. If f(x) approaches a finite number L as x approaches a (x ≠ a), we say lim(x→a) f(x) = L.
2. Symbols
SymbolMeaning
f(x)The function under study
aThe point x approaches
LThe limit (exists iff left and right limits agree)
hStep, shrunk successively to approach a
3. How this tool works
  • Take h = 0.1, 0.01, 0.001 and compute f(a−h) and f(a+h);
  • If both sides settle to the same constant L, the limit exists;
  • If the two sides approach different constants (or diverge), the limit does not exist;
  • Numerical approach is an illustration; the rigorous proof uses the epsilon-delta language.
4. Steps
  1. Enter an expression, e.g. (x^2-1)/(x-1);
  2. Enter the approach point a = 1;
  3. Click Calculate and watch f(1−h), f(1+h) as h shrinks;
  4. Click Load Example to fill the sample instantly.
5. Example
Example: Find lim(x→1) (x²−1)/(x−1).
Solution: (x²−1)/(x−1)=x+1 (x≠1), so as x→1 the limit is 2. With h=0.001, f(0.999)≈1.999 and f(1.001)≈2.001, both approaching 2.
6. Pitfalls
x→a means x ≠ a; f may be undefined at a yet still have a limit;
The left and right limits must agree for the limit to exist;
Numerical approach has rounding error; very small h can give floating-point noise.

❓ FAQ | Definition of a Limit

When does a limit exist?
When both the left-hand and right-hand limits exist and are equal. A single-sided approach, or two different values, means no limit.
If f(a) is undefined, can the limit still exist?
Yes. The limit only cares about the trend as x gets near a, not about f(a) itself. For example (x²−1)/(x−1) is undefined at x=1 but its limit is 2.
Why numerical approach instead of an exact solver?
This tool helps you build intuition by shrinking h and watching the trend; a rigorous proof needs the epsilon-delta definition or algebraic simplification.
Which expressions are supported?
+ − * / power ^ parentheses, plus sin, cos, tan, sqrt, exp, log, and the constants pi and e. Never omit the multiplication sign: write 2*x, not 2x.
Why is the result slightly off my hand calculation?
Floating-point precision may add noise in the last digits when h is tiny; this is normal and does not change the limit value.
What if left and right limits differ?
It is a jump discontinuity at that point, so lim(x→a) f(x) does not exist, e.g. a piecewise function taking different constants on each side.