I know all the steps of how they worked the limit out, but I am having trouble interpreting its meaning. If I substituent 1 into my equation I get 3. And this is as close to $\sqrt5$ $\approx$ 2.23 as I can get. But even this feels a bit far. I mean why couldn't the limit be 2.5 or 2.99?
As an example, the value of $\frac{sin(n)}{n}$ is very close to 0 for large values of n.