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Limit Calculator

Evaluate one-sided and two-sided limits with live numerical analysis.

Limit Setup

Pick a function and the point to approach.

minimum h is 0.0001
Numerical Approach
Enter a valid approach point.
Evaluate one-sided and two-sided limits with step-by-step analysis.

About this topic

Limit Definition The value a function approaches as input approaches a point.

A limit describes what value f(x) approaches as x gets arbitrarily close to a point a — even if f(a) itself is undefined.

Key formulas & rules

  • •lim(x→a) f(x) = L means f(x) → L as x → a
  • •Left limit: lim(x→a⁻) — approach from below
  • •Right limit: lim(x→a⁺) — approach from above
  • •Limit exists ⇔ left limit = right limit
  • •Function need not be defined at a
  • •Infinity is a valid limit value

Examples

Input Output Note
lim(x→2) (x²−4)/(x−2) 4 Removable hole
lim(x→0) 1/x DNE Left ≠ Right
lim(x→∞) 1/x 0 Approaches 0
lim(x→3) x² 9 Continuous

Real-world uses

Derivative foundation Continuity analysis Asymptote detection
One-Sided Limits Approach the point from only one direction.

One-sided limits are essential when the function behaves differently on each side of the point, or is only defined on one side.

Key formulas & rules

  • •Left: x approaches a from below (x < a)
  • •Right: x approaches a from above (x > a)
  • •Notation: lim(x→a⁻) and lim(x→a⁺)
  • •Both equal → two-sided limit exists
  • •Jump discontinuity: left ≠ right
  • •√x has only a right limit at x = 0

Examples

Input Output Note
lim(x→0⁻) 1/x −∞ From below
lim(x→0⁺) 1/x +∞ From above
lim(x→0⁺) √x 0 One-sided only
lim(x→0) |x|/x DNE Jump

Real-world uses

Piecewise functions Signal jumps Domain boundaries
L'Hôpital's Rule Differentiate top and bottom for 0/0 or ∞/∞ forms.

When a limit is an indeterminate form (0/0 or ∞/∞), differentiate the numerator and denominator separately and take the limit again.

Key formulas & rules

  • •Applies to 0/0 and ∞/∞ forms
  • •lim f/g = lim f'/g'
  • •Can be applied repeatedly
  • •Not valid for other indeterminate forms directly
  • •Other forms (0·∞, ∞−∞, 1^∞) need rewriting
  • •Example: lim sin(x)/x → lim cos(x)/1 = 1

Examples

Input Output Note
lim(x→0) sin(x)/x 1 Apply L'Hôpital
lim(x→∞) (eˣ)/(x²) ∞ Exponential wins
lim(x→0) (1−cos x)/x² ½ Apply twice
lim(x→∞) ln(x)/x 0 Log loses to polynomial

Real-world uses

Computing derivatives Series convergence Physics asymptotics
Squeeze Theorem Bound a tricky function between two known ones.

If g(x) ≤ f(x) ≤ h(x) near a, and lim g = lim h = L, then lim f = L too. Powerful for oscillating functions.

Key formulas & rules

  • •Need g(x) ≤ f(x) ≤ h(x) near a
  • •lim g(x) = lim h(x) = L
  • •Then lim f(x) = L
  • •Common with sin(1/x), x·sin(1/x)
  • •Works for one-sided and infinite limits
  • •Also called Sandwich Theorem

Examples

Input Output Note
lim(x→0) x·sin(1/x) 0 Squeeze with ±x
lim(x→0) x²·cos(1/x) 0 Squeeze with ±x²
lim(x→∞) sin(x)/x 0 Squeeze with ±1/x
lim(x→0) x·cos(1/x²) 0 Oscillating

Real-world uses

Oscillating signals Convergence proofs Bound analysis

Modes & formulas

1

Numeric Limit Evaluation

• Choose a preset function or enter a custom one. • Enter the approach point (including ±∞). • The calculator samples the function from the left and right, builds an approach table, and estimates the limit.
lim(x→a) f(x) = L

If both the left limit and right limit converge to the same value L, the two-sided limit exists and equals L. If they differ, the limit does not exist (DNE).

2

Left / Right / Both

• Choose which side(s) to approach from. • The table shows sample values from each side with decreasing step sizes. • The tool detects convergence when successive values differ by less than 10⁻⁴.
lim(x→a⁻) f(x) = L₁ | lim(x→a⁺) f(x) = L₂

If L₁ = L₂, the two-sided limit equals that value. If L₁ ≠ L₂, the limit does not exist. This is common in piecewise functions.

How to use it

Pick a preset function, enter the approach point (e.g., 0, 1, inf, ∞), and set a step size. The table and graph show the function approaching the limit point.

Frequently asked questions

What does DNE mean?
DNE stands for 'Does Not Exist.' A limit DNE when the left and right limits disagree, or when the function diverges to ±∞ on one or both sides.
Can a limit exist where the function is undefined?
Yes. The classic example is sin(x)/x at x = 0: the function is undefined at 0, but the limit is 1.
What is a limit?
A limit describes the value a function approaches as the input gets closer and closer to a certain point.
What is a one-sided limit?
A limit where you only approach the target point from one direction (either from values smaller than the target, or larger).
How do you evaluate a limit numerically?
By plugging in values extremely close to the target point and observing the trend of the outputs. The calculator does this automatically.
Can a limit exist if the function is undefined at that point?
Yes. The classic example is sin(x)/x at x=0. The function has a "hole" at 0, but the limit as x approaches 0 is 1.

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Read the full Limit Calculator guide — examples & 18 FAQs