Limit as x -> infinity of (8x^3 - 1)/(2x^3 + x)

Step by step

The question

Question: find the limit as x → ∞ of (8x³ − 1)/(2x³ + x).

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 8x³.

At the bottom it is 2x³.

See it with x = 1000.

8x³ is 8,000,000,000.

The other part is tiny next to that.

2 Keep only those

So we keep only 8x³ over 2x³.

x³ over x³ is 1.

Cross them out.

Only the numbers are left: 8/2.

They do not change when x grows.

8 divided by 2 is 4.

So the limit is 4.

3 Check

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

A calculator gives about 4.

With x = 10000 it is even closer to 4.

Right!

Let's look back.

First: find the highest powers.

Then: keep only those.

Last: check.