Worked examples
The question
Question: find the limit as x → ∞ of (3x² + 1)/(x² − 4).
lim x → ∞ 3x2 + 1x2 − 4 =
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 x².
See it with x = 1000.
3x² is 3,000,000.
The other part is tiny next to that.
x = 1000: 3x² = 3,000,000
1: tiny next to it
2 Keep only those
So we keep only 3x² over x².
= lim x → ∞ 3x²x²
x² over x² is 1.
Cross them out.
Careful!
Not 1 over −4, the plain numbers!
They matter when x is near 0.
Not when x is huge.
Careful: not 1−4 ✗
Only the number is left: 3.
It does not change when x grows.
So the limit is 3.
= 3
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 3.
x = 1000: f(x) ≈ 3.0000
With x = 10000 it is even closer to 3.
Right!
x = 10000: f(x) ≈ 3.0000
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
▶ Watch it solved
The question
Question: find the limit as x → ∞ of (2x + 1)/(x² + 3).
lim x → ∞ 2x + 1x2 + 3 =
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 2x.
At the bottom it is x².
See it with x = 1000.
2x is 2,000.
The other part is tiny next to that.
x = 1000: 2x = 2,000
1: tiny next to it
2 Keep only those
So we keep only 2x/x².
= lim x → ∞ 2xx × x
Split the bigger power: x² is x × x.
x² = x × x
x/x is 1.
Cross them out.
Write what is left.
x is still at the bottom.
= lim x → ∞ 2x
2/a huge number is tiny.
As x grows, it gets closer and closer to 0.
x = 1000: 2 ÷ 1,000 = 0.002
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 0.002.
x = 1000: f(x) ≈ 0.0020
With x = 10000 it is even closer to 0.
Right!
x = 10000: f(x) ≈ 0.0002
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
▶ Watch it solved
The question
Question: find the limit as x → ∞ of (x³ − 1)/(2x² + 5).
lim x → ∞ x3 − 12x2 + 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.
x = 1000: x³ = 1,000,000,000
−1: tiny next to it
2 Keep only those
So we keep only x³ over 2x².
= lim x → ∞ x² × x2x²
Split the bigger power: x³ is x² × x.
x³ = x² × x
x² over x² is 1.
Cross them out.
Write what is left.
x is still on top.
= lim x → ∞ 12x
When x is huge, this is huge too.
It grows without end.
x = 1000: x = 1,000
Careful!
The top grows faster than the bottom.
So the fraction never settles on a number.
Careful: top power bigger → no 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.
x = 1000: f(x) ≈ 500
With x = 10000 it is about 5000.
It keeps growing.
Right!
x = 10000: f(x) ≈ 5000
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
▶ Watch it solved
