We have `x sin x=1`
or `sin x=1/x`
To find the number of roots of above equation, we draw the graphs of `y= sin x` and `y =1/x` and count the number of points of intersection.
Now for `y=1/x`.
we have `1/x rarr oo`, when `x rarr 0^(+)`
On increasing the value of x from `0^(+)` onwards, value of `1/x` decreases.
Also, `1/x rarr 0^(+)`, when `x rarr oo`
Similarly, `1/x rarr -oo`, when `x rarr 0^(-)`
On decreasing the value of x from `0^(-)`, value of `1/x` increases.
Thus graphs of `y=1/x` and `y sin x` are as shown in the following figure.
From the figure, we can observe that both graphs will intersect each other infinite number of times.
Hence, equation `sin x=1/x or x sin x=1` has infinite roots.
