I'm trying to find what this function converges to.
$$ \sqrt{n} \sin(π/\sqrt{n}) $$
I've taken the limit:
$$ \lim_{n\rightarrow\infty} \sqrt{n}\sin(π/\sqrt{n}) $$
It appears as though the function takes the following form as n approaches infinity:
\begin{align*}
\lim_{n\rightarrow\infty} \infty \sin(π/\infty) &= \lim_{n\rightarrow\infty} \infty\cdot \sin(0)\\
&= \infty
\end{align*}
I have been told that this converges to $\pi$, which means I'm doing something wrong here. What am I doing wrong?
Best Answer
Recall that $$\lim_{x \to 0} \dfrac{\sin(ax)}{x} = a$$