Limit as x -> infinity of (5x + 2)/(x + 9)

Step by step

The question

Question: find the limit as x → ∞ of (5x + 2)/(x + 9).

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 5x.

At the bottom it is x.

See it with x = 1000.

5x is 5,000.

The other part is tiny next to that.

2 Keep only those

So we keep only 5x over x.

x/x is 1.

Cross them out.

Careful!

Not 2/9, the plain numbers!

They matter when x is near 0.

Not when x is huge.

Only the number is left: 5.

It does not change when x grows.

So the limit is 5.

3 Check

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

A calculator gives about 4.9574.

With x = 10000 it is even closer to 5.

Right!

Let's look back.

First: find the highest powers.

Then: keep only those.

Last: check.