Step by step
The question
Question: find the limit as x → ∞ of (5x³ − 2x)/(2x³ + 7).
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 5x³.
At the bottom it is 2x³.
See it with x = 1000.
5x³ is 5,000,000,000.
The other part is tiny next to that.
2 Keep only those
So we keep only 5x³ over 2x³.
x³ over x³ is 1.
Cross them out.
Only the numbers are left: 5/2.
They do not change when x grows.
So the limit is 5/2.
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 2.5.
With x = 10000 it is even closer to 5/2.
Right!
4 Pick the choice
Our answer is limit = 5/2.
That is choice D.
5 Why not the others
A is 0.
Wrong: the top and the bottom have the same highest power.
B is infinity.
Wrong: the top does not grow faster than the bottom.
C is 5/7.
Wrong: used 5 and 7: use the numbers in front of the highest powers.
Let's look back.
First: find the highest powers.
Then: keep only those.
Then: check.
Then: pick the choice.
Last: why not the others.