Step by step
The question
Question: find the limit as x → ∞ of (6x − 1)/(3x + 2).
lim x → ∞ 6x − 13x + 2 =
1 Find the highest powers
x goes to infinity: x gets huge, bigger than any number.
Then the highest power wins.
On top it is 6x.
At the bottom it is 3x.
See it with x = 1000.
6x is 6,000.
The other part is tiny next to that.
x = 1000: 6x = 6,000
−1: tiny next to it
2 Keep only those
So we keep only 6x/3x.
= lim x → ∞ 6x3x
x/x is 1.
Cross them out.
Careful!
Not −1/2, the plain numbers!
They matter when x is near 0.
Not when x is huge.
Careful: not −12 ✗
Only the numbers are left: 6/3.
They do not change when x grows.
= 63
6 divided by 3 is 2.
So the limit is 2.
= 2
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 1.9983.
x = 1000: f(x) ≈ 1.9983
With x = 10000 it is even closer to 2.
Right!
x = 10000: f(x) ≈ 1.9998
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
Answer
= 2