Graph of sin x / x (the sinc function)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of y = sin x / x is a wave that dies away in both directions. At x = 0 the formula is undefined, 0/0, but the graph heads smoothly to 1, the most famous limit in calculus.

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The graph of

y=sin(x)x

for −20 ≤ x ≤ 20. Dots mark its roots, turning points and y-intercept.

Key features

Domain: x ≠ 0 (the gap is removable; sinc 0 is defined as 1)
Range: About −0.217 ≤ y < 1
Limit at 0: lim x→0 sin x / x = 1
Zeros: x = nπ, n ≠ 0
Envelope: −1/|x| ≤ y ≤ 1/|x|
Symmetry: Even
Total area: ∫ sin x / x dx over all reals = π
y-intercept: None strictly (undefined at 0), but the limit there is 1
Derivative: −sin(x)x²+cos(x)x
Antiderivative: No elementary form; it is the sine integral Si(x) + C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of sin x / x looks like

Dividing sin x by x shrinks each wave by its distance from the origin, so the oscillations decay like 1/x. The first negative dip, near x = 4.49, reaches about −0.217, the lowest value of the function.

At x = 0 both numerator and denominator are zero. The limit is 1 because sin x ≈ x for small x, and the calculator shows a hole there because the formula itself is undefined. Defining sinc 0 = 1 fills it in and makes the function smooth everywhere.

The function has no elementary antiderivative, yet its total signed area is exactly π, a classic result known as the Dirichlet integral.

Points of interest

Found numerically to about twelve digits in the window −20 ≤ x ≤ 20, then shown in exact form where one exists.

x-intercepts (roots): x = −18.8496, x = −15.708, x = −12.5664, x = −9.4248, x = −6.2832, x = −3.1416 … (12 in view)
Local maxima: (−14.0662, 0.0709), (−7.7253, 0.1284), (7.7253, 0.1284), (14.0662, 0.0709)
Local minima: (−17.2208, −0.058), (−10.9041, −0.0913), (−4.4934, −0.2172), (4.4934, −0.2172), (10.9041, −0.0913), (17.2208, −0.058)
Inflection points: (−18.7426, −0.00569326), (−15.5792, 0.00823992), (−12.4044, −0.0129969), (−9.20584, 0.0235929), (−5.94037, −0.0565857), (−2.08158, 0.419088) … (12 in view)

The calculator does not mark a y-intercept because sin x / x is genuinely undefined at x = 0; it draws the curve up to the hole on both sides.

Table of values

A few points to plot, with exact values where they are simple.

Values of sin x / x
x	y = sin x / x
−π	0
−π/2	0.63662
0.01	0.999983
π/6	0.95493
π/2	0.63662
π	0
2π	0

How to graph sin x / x by hand

1
Plot the limit point (0, 1) as an open circle, since the formula is undefined there.
2
Mark the zeros at ±π, ±2π, ±3π.
3
Draw the envelope curves y = ±1/x lightly and fit decaying waves between them.

Transformations

How the graph moves when you change the formula. The original is sin x / x in blue.

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y = 1/x The upper envelope for x > 0.
y = -1/x The lower envelope for x > 0.
y = (sin(x)/x)^2 The squared sinc, the diffraction pattern of a single slit.

Where it shows up

Signal processing: the ideal low-pass filter and the Fourier transform of a rectangular pulse.
Optics: the diffraction pattern of light through a single slit is proportional to sinc².