Step by step
The question
Question: find the limit as x → ∞ of (3x² + 1)/(x³ − 4).
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 3x².
At the bottom it is x³.
See it with x = 1000.
3x² is 3,000,000.
The other part is tiny next to that.
2 Keep only those
So we keep only 3x² over x³.
Split the bigger power: x³ is x² × x.
So we write x³ as x² × x.
x² over x² is 1.
Cross them out.
We write it again, without the crossed-out parts.
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!
4 Pick the choice
Our answer is limit = 0.
That is choice A.
5 Why not the others
B is 3.
Wrong: the bottom has the higher power, x³.
C is −1/4.
Wrong: used the numbers 1 and −4.
D is infinity.
Wrong: the bottom grows faster than the top.
Let's look back.
First: find the highest powers.
Then: keep only those.
Then: check.
Then: pick the choice.
Last: why not the others.