Step by step
The question
Question: find the limit as x → ∞ of (x² + 5)/(2x² − 3).
lim x → ∞ x2 + 52x2 − 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 x².
At the bottom it is 2x².
See it with x = 1000.
x² is 1,000,000.
The other part is tiny next to that.
x = 1000: x² = 1,000,000
5: tiny next to it
2 Keep only those
So we keep only x² over 2x².
= lim x → ∞ x²2x²
x² over x² is 1.
Cross them out.
Careful!
Not 5 over −3, the plain numbers!
They matter when x is near 0.
Not when x is huge.
Careful: not 5−3 ✗
Only the numbers are left: 1/2.
They do not change when x grows.
So the limit is 1/2.
= 12
3 Check
Let's check: put x = 1000 into the whole fraction.
A calculator gives about 0.5.
x = 1000: f(x) ≈ 0.5000
With x = 10000 it is even closer to 1/2.
Right!
x = 10000: f(x) ≈ 0.5000
Let's look back.
First: find the highest powers.
Then: keep only those.
Last: check.
Answer
= 12