Limit as x -> infinity of (3x^2 + 1)/(x^2 - 4)

Step by step

The question

Question: find the limit as x → ∞ of (3x² + 1)/(x² − 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.

2 Keep only those

So we keep only 3x² over x².

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.

Only the number is left: 3.

It does not change when x grows.

So the limit is 3.

3 Check

Let's check: put x = 1000 into the whole fraction.

A calculator gives about 3.

With x = 10000 it is even closer to 3.

Right!

Let's look back.

First: find the highest powers.

Then: keep only those.

Last: check.