Step by step
The question
Question: find the limit as x → ∞ of (4x + 1)/(x³ − 2).
lim x → ∞ 4x + 1x3 − 2 =
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 4x.
At the bottom it is x³.
See it with x = 1000.
4x is 4,000.
The other part is tiny next to that.
x = 1000: 4x = 4,000
1: tiny next to it
2 Keep only those
So we keep only 4x/x³.
= lim x → ∞ 4xx × x²
Split the bigger power: x³ is x × x².
x³ = x × x²
x/x is 1.
Cross them out.
Write what is left.
x² is still at the bottom.
= lim x → ∞ 4x²
4/a huge number is tiny.
As x grows, it gets closer and closer to 0.
x = 1000: 4 ÷ 1,000,000 = 0.000004
Careful!
The bottom grows faster than the top.
So the fraction gets smaller and smaller.
Careful: top power smaller → 0
So the limit is 0.
= 0
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 0.
x = 1000: f(x) ≈ 0.0000
With x = 10000 it is even closer to 0.
Right!
x = 10000: f(x) ≈ 0.0000
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
Answer
= 0