Graph of the Sinc Function $y = \dfrac{\sin x}{x}$

The function $y = \dfrac{\sin x}{x}$ is known as the sinc function. Here we treat the unnormalized form.

Definition and singularity

Formally the denominator vanishes at $x = 0$, but the limit

exists, so $x = 0$ is a removable singularity. Defining the value there to be $1$ makes the graph continuous, passing through the point $(0,\ 1)$.

Domain, range, and symmetry

With the value at $x = 0$ filled in as above, the domain is all real numbers. Since both $\sin x$ and $x$ are odd, their quotient is even, so the graph is symmetric about the $y$-axis. Its maximum value is $1$, attained at $x = 0$.

Behavior and limits

As $x \to \pm\infty$, the bounded numerator $|\sin x| \le 1$ is divided by the ever-larger $|x|$, so $\dfrac{\sin x}{x} \to 0$. The oscillation decays within the envelope $\pm\dfrac{1}{x}$, shrinking toward zero.

Zeros

The function is zero where $\sin x = 0$ with $x \ne 0$, that is at $x = n\pi$ for $n \ne 0$ (for example $\pm\pi,\ \pm 2\pi,\ \dots$). Note that $x = 0$ is not a zero but rather the point of maximum value.

Extrema and specific values

One local extremum appears between each pair of adjacent zeros, shrinking along the envelope $\dfrac{1}{|x|}$ as $x$ moves away from the origin. For example $\dfrac{\sin(\pi/2)}{\pi/2} = \dfrac{2}{\pi} \approx 0.64$. The first trough after the origin (near $x \approx 4.49$) reaches about $-0.217$, so the values always lie between roughly $-0.217$ and $1$.

Relation to other functions and applications

The sinc function is the Fourier transform of a rectangular pulse and plays a central role in signal processing, Fourier analysis, and the study of diffraction. In the sampling theorem it is the interpolation kernel — an ideal low-pass filter — used to reconstruct a continuous signal from discrete samples. The form $\dfrac{\sin(\pi x)}{\pi x}$ is the normalized sinc, adjusted so that its zeros fall exactly on the integers.