lim_(xrarr1)sin(π/(x-1))=?

lim_(xrarr1)sin(π/(x-1))

Dec 12, 2017

The limit does not exist.

Explanation:

As $x$ approaches $1$, the argument, $\frac{\pi}{x - 1}$ takes on values $\frac{\pi}{2} + 2 \pi k$ and $\frac{3 \pi}{2} + 2 \pi k$ infinitely often.

So $\sin \left(\frac{\pi}{x - 1}\right)$ takes on values $- 1$ and $1$, infinitely many times.

The value cannot be approaching a single limiting number.

graph{sin(pi/(x-1)) [-1.796, 8.07, -1.994, 2.94]}