LIM without x->a

Question

I was working through the first few pages of Problem-Solving Strategies by Arthur Engel (Which may or may not be a little above my level), and I came upon an interesting form of notation I haven't seen before:

$$ lim \: x_n = lim \: y_n = x $$

Is this a legitimate form of notation or one of the many errata supposedly present in the book?

Answer 1

$x_n$ and $y_n$ are sequences rather than functions. With a function you might ask "limit as the function goes where?" but with a sequence, the answer to that question is always, "as we get further and further along in the sequence". That is, as $n$ (the term of the sequence), approaches infinity.

In this case, the notation says that the sequence $x_n$ and the sequence $y_n$ both approach the same number $x$.

Answer 2

It is legitimate. Without context it is probably the limit of a real sequence - $\lim_{n\to \infty}x_n$.