Step by step
The question
Question: find the limit as x → ∞ of (3x − 4)/(x² + 1).
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 3x.
At the bottom it is x².
See it with x = 1000.
3x is 3,000.
The other part is tiny next to that.
2 Keep only those
So we keep only 3x/x².
Split the bigger power: x² is x × x.
x/x is 1.
Cross them out.
Write what is left.
x is still at the bottom.
3/a huge number is tiny.
As x grows, it gets closer and closer to 0.
Careful!
The bottom grows faster than the top.
So the fraction gets smaller and smaller.
So the limit is 0.
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 0.003.
With x = 10000 it is even closer to 0.
Right!
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.