Step by step
The question
Question: find the limit as x → ∞ of (x³ − 1)/(2x² + 5).
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 x³.
At the bottom it is 2x².
See it with x = 1000.
x³ is 1,000,000,000.
The other part is tiny next to that.
2 Keep only those
So we keep only x³ over 2x².
Split the bigger power: x³ is x² × x.
x² over x² is 1.
Cross them out.
Write what is left.
x is still on top.
When x is huge, this is huge too.
It grows without end.
Careful!
The top grows faster than the bottom.
So the fraction never settles on a number.
It grows without end.
We write ∞: there is no number it gets close to.
3 Check
Let's check: put x = 1000 into the whole fraction.
We get about 500.
With x = 10000 it is about 5000.
It keeps growing.
Right!
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.