Why does this this limit not exist?
$$\lim_{x \to \infty} \arcsin \left({x+1\over x}\right)$$
According to me on dividing both the numerator and the denominator by $x$ and then putting $ x = \infty $ we should get $ \arcsin (1) $ which is equal to $ \frac{\pi} 2$ . Where am I wrong?
Best Answer
Because $\frac{x+1}{x}>1$ for $x>0$, and $\arcsin{y}$ is not defined for $y>1$.
On the other hand, the limit as $x \to -\infty$ does exist, since $-1<\frac{x+1}{x}<1$ for sufficiently large negative $x$, and is $\pi/2$.