Step by step
The question
Question: find the limit as x → ∞ of 5/(x² + 1).
lim x → ∞ 5x2 + 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 5.
At the bottom it is x².
2 Keep only those
So we keep only 5/x².
= lim x → ∞ 5x⁰x⁰ × x²
Split the bigger power: x² is x⁰ × x².
x² = x⁰ × x²
x⁰ over x⁰ is 1.
Cross them out.
Write what is left.
x² is still at the bottom.
= lim x → ∞ 5x²
5/a huge number is tiny.
As x grows, it gets closer and closer to 0.
x = 1000: 5 ÷ 1,000,000 = 0.000005
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