Limit as x -> infinity of (9x^2 - 4)/(3x^2 + 2x)

Step by step

The question

Question: find the limit as x → ∞ of (9x² − 4)/(3x² + 2x).

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 9x².

At the bottom it is 3x².

See it with x = 1000.

9x² is 9,000,000.

The other part is tiny next to that.

2 Keep only those

So we keep only 9x² over 3x².

x² over x² is 1.

Cross them out.

Only the numbers are left: 9/3.

They do not change when x grows.

9 divided by 3 is 3.

So the limit is 3.

3 Check

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

A calculator gives about 2.998.

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.