Step by step
The question
Question: find the limit as x → ∞ of (8x³ − 1)/(2x³ + x).
What happens to f(x) when x gets huge?
There are four choices.
First we solve it, the normal way.
Then we pick.
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 8x³.
At the bottom it is 2x³.
See it with x = 1000.
8x³ is 8,000,000,000.
The other part is tiny next to that.
2 Keep only those
So we keep only 8x³ over 2x³.
x³ over x³ is 1.
Cross them out.
Only the numbers are left: 8/2.
They do not change when x grows.
8 divided by 2 is 4.
So the limit is 4.
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 4.
With x = 10000 it is even closer to 4.
Right!
4 Pick the choice
Our answer is limit = 4.
That is choice B.
5 Why not the others
A is 0.
Wrong: the top and the bottom have the same highest power.
C is −1.
Wrong: used −1 and x: use the numbers in front of x³.
D is 8.
Wrong: lost the 2 at the bottom.
Let's look back.
First: find the highest powers.
Then: keep only those.
Then: check.
Then: pick the choice.
Last: why not the others.