Step by step
The question
Question: find the limit of the sequence a n = (n² + 1)/n, as n grows.
1 The first terms
Work out a few terms: n = 1, 2, 3, then 10 and 100.
Put each n in.
Look at the terms.
They get bigger and bigger.
2 Divide by the highest power
Divide every term, top and bottom, by n, the highest power at the bottom.
The value does not change.
As n grows, 1/n gets as small as we like: it goes to 0.
So does any number over a power of n.
3 The limit
Every part with n at the bottom goes to 0.
The top still has n in it: it grows without end.
No limit.
4 Decide
There is no single number at the end: the sequence diverges.
Careful!
Converges means the terms settle on one number.
5 Check
Let's check with a calculator: term number 1000 is about 1000.001.
Still growing.
Right!
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.