Step by step
The question
Question: find the limit of the sequence a n = (6n + 1)/(3n² + 2), as n grows.
an = 6n + 13n2 + 2
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 0.
They get close to 0
2 Divide by the highest power
Divide every term, top and bottom, by n to the 2, the highest power at the bottom.
The value does not change.
an = 6n + 1n23 + 2n2
As n grows, 1/n gets as small as we like: it goes to 0.
So does any number over a power of n.
1n → 0
3 The limit
Every part with n at the bottom goes to 0.
as n → ∞ = 0 + 03 + 0
What is left: 0.
= 0
4 Decide
The terms settle on one number: the sequence converges to 0.
= converges to 0
Careful!
Converges means the terms settle on one number.
Careful: a limit is one number the terms settle on
5 Check
Let's check with a calculator: term number 1000 is about 0.002.
Close to 0.
Right!
n = 1000: an ≈ 0.002
Let's look back.
First: the first terms.
Then: divide by the highest power.
Then: the limit.
Then: decide.
Last: check.
Answer
= converges to 0