# How do you graph y=(sinx)/x?

Apr 19, 2015

We know that the limit in 0 is 1

(it's one of the notables limits: in a neighbourhood of 0 $\sin \left(x\right) = x + o \left({x}^{2}\right) \implies \sin \frac{x}{x} = 1 + o \left(x\right) \to 1 \mathmr{if} x \to 0$ )

We know it is an even function (quotient of two odd functions), so the graph must be symmetric.

We concentrate on $x > 0$, and then extend by symmetry

We know it has zeros where $\sin \left(x\right)$ has zeros (except for $x = 0$) so it has zeros in $x = k \pi , k \ne 0$.

Then we know that $\sin \left(\frac{\pi}{2} + 2 k \pi\right) = 1$, so we know that the function in that points is like $\frac{1}{x}$

For the same reason in $x = \frac{3 \pi}{2} + 2 k \pi$ it is like $- \frac{1}{x}$

So you draws the four branches of hyperboles and consider the incidence in the points, consider the zeros, consider than in 0 is 1 and consider the symmetry.

graph{1/x [-10, 10, -5, 5]}
graph{-1/x [-10, 10, -5, 5]}
graph{sinx/x [-10, 10, -5, 5]}