Step by step
The question
Question: find the limit as x → ∞ of (6 − x²) over (2x² + 1).
lim x → ∞ 6 − x22x2 + 1 =
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
6: 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 6/1, the plain numbers!
They matter when x is near 0.
Not when x is huge.
Careful: not 61 ✗
Only the numbers are left: −1/2.
They do not change when x grows.
= −12
In lowest terms that is −1/2.
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