Limit of (x^2 - 6x + 9)/(x - 3) as x -> 3 using a table Step by Step

Step by step

The question

Question: find the limit as x → 3 of (x² − 6x + 9)/(x − 3), using a table.

lim x → 3 x² − 6x + 9x − 3 =

1 Try x = 3

First, try x = 3 itself.

The top is 0, and the bottom is 0.

3² − 6(3) + 93 − 3 = 00

We cannot divide by 0.

So f(x) has no value at 3.

00: no value

But a limit does not need the value at 3.

It asks: as x gets close to 3, what number does f(x) get close to?

limit: the number f(x) gets close to

Careful!

0/0 does not mean there is no limit.

It only means we must look closer.

Careful: not 00 → no limit ✗

2 From the left

Take numbers just below 3: 2.9, 2.99, 2.999.

They come closer and closer to 3.

Put 2.9 in: we get −0.1.

f(2.9) = (2.9)² − 6(2.9) + 9(2.9) − 3 = 8.41 − 17.4 + 92.9 − 3 = 0.01−0.1 = −0.1

−0.1

The others the same way, with a calculator.

−0.01

−0.001

From the left, f(x) gets close to 0.

from the left: f(x) → 0

3 From the right

Now numbers just above 3: 3.1, 3.01, 3.001.

Work out f(x) for each one.

0.1

0.01

0.001

From the right, f(x) gets close to 0 too.

from the right: f(x) → 0

4 Read the limit

Both sides get close to the same number.

So the limit is 0.

lim x → 3 x² − 6x + 9x − 3 = 0

5 Check

Let's check with numbers even closer.

x = 2.9999 gives −0.0001.

x = 2.9999: f(x) = −0.0001

x = 3.0001 gives 0.0001.

Still close to 0.

Right!

x = 3.0001: f(x) = 0.0001

Let's look back.

First: try x = 3.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer