Step by step
The question
Question: find the limit of the sequence a n = (n³ − 2n)/(4n³ + 1), 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 close to 1/4.
2 Divide by the highest power
Divide every term, top and bottom, by n to the 3, 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.
What is left: 1/4.
4 Decide
The terms settle on one number: the sequence converges to 1/4.
Careful!
Top and bottom both grow, but that does not make the limit 1.
Divide by the highest power and see what is left.
5 Check
Let's check with a calculator: term number 1000 is about 0.25.
Close to 1/4.
Right!
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.