Meaning of divergent limit? $$\lim_{x\rightarrow \infty} {\frac{1}{\sin x}}$$

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*What is its value?

*What is the meaning of limit diverges?

*Can I say here that the limit does not exist?


It does not converge to any value, it diverges by oscillation.
1/sin(x)=csc(x) and the csc keeps cyclically jumping from -inf -> -1 -> -inf -> inf -> 1 -> inf -> -inf ->...... and never converges to anything. This is like asking what is the limit of the sequence 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1..... as the index tends to infinity.
