Limit as x -> infinity of (2x^2 + 3x)/(5x^2 - 1)

Step by step

The question

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

At the bottom it is 5x².

See it with x = 1000.

2x² is 2,000,000.

The other part is tiny next to that.

2 Keep only those

So we keep only 2x² over 5x².

x² over x² is 1.

Cross them out.

Only the numbers are left: 2/5.

They do not change when x grows.

So the limit is 2/5.

3 Check

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

A calculator gives about 0.4006.

With x = 10000 it is even closer to 2/5.

Right!

Let's look back.

First: find the highest powers.

Then: keep only those.

Last: check.