$lim_(xrarr1)sin(π/(x-1))$ $=$ ?

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$lim_(xrarr1)sin(π/(x-1))$

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Answer 1 · 2017-12-12T14:48:57

The limit does not exist.

As $x$ approaches $1$ , the argument, $pi/(x-1)$ takes on values $pi/2 + 2pik$ and $(3pi)/2 + 2pik$ infinitely often.

So $sin(pi/(x-1))$ 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]}