Which of the following two examples is the correct use of the $\rightarrow$ "tends to" relation?
Example 1
$$ \lim_{n \rightarrow \infty} \frac{\pi(n)}{n/\ln(n)} = 1 $$
Example 2 $$ \lim_{n \rightarrow \infty} \frac{\pi(n)}{n/\ln(n)} \rightarrow 1 $$
If both are valid, or conventional, please do state in which contexts the traditions exist.
The correct notation is the first one. The second is wrong since a limit cannot “tend” to anything. The limit is a real number (in this case).
The limit equals the value, while the sequence of numbers tends to the value. We say that a sequence of numbers “tends” to a number when we mean that its limit “equals” 1.
Use example $1$, as the limit is a number by definition. However, there is another option: $\tfrac{\pi(n)}{n/\ln n}\to1$ as $n\to\infty$. These clauses may be listed in the other order, especially when describing several $n\to\infty$ limits.
