Limit as x -> infinity of (4x^2 - x)/(2x^2 + 7)

Step by step

The question

Question: find the limit as x → ∞ of (4x² − x)/(2x² + 7).

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 4x².

At the bottom it is 2x².

See it with x = 1000.

4x² is 4,000,000.

The other part is tiny next to that.

2 Keep only those

So we keep only 4x² over 2x².

x² over x² is 1.

Cross them out.

Only the numbers are left: 4/2.

They do not change when x grows.

4 divided by 2 is 2.

So the limit is 2.

3 Check

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

A calculator gives about 1.9995.

With x = 10000 it is even closer to 2.

Right!

Let's look back.

First: find the highest powers.

Then: keep only those.

Last: check.