Limit of (x^2 - 7x + 10)/(x - 2) as x -> 2 using a table Step by Step

Step by step

The question

Question: find the limit as x → 2 of (x² − 7x + 10)/(x − 2), using a table.

lim x → 2 x² − 7x + 10x − 2 =

1 Try x = 2

First, try x = 2 itself.

The top is 0, and the bottom is 0.

2² − 7(2) + 102 − 2 = 00

We cannot divide by 0.

So f(x) has no value at 2.

00: no value

But a limit does not need the value at 2.

It asks: as x gets close to 2, 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 2: 1.9, 1.99, 1.999.

They come closer and closer to 2.

Put 1.9 in: we get −3.1.

f(1.9) = (1.9)² − 7(1.9) + 10(1.9) − 2 = 3.61 − 13.3 + 101.9 − 2 = 0.31−0.1 = −3.1

−3.1

The others the same way, with a calculator.

−3.01

−3.001

From the left, f(x) gets close to −3.

from the left: f(x) → −3

3 From the right

Now numbers just above 2: 2.1, 2.01, 2.001.

Work out f(x) for each one.

−2.9

−2.99

−2.999

From the right, f(x) gets close to −3 too.

from the right: f(x) → −3

4 Read the limit

Both sides get close to the same number.

So the limit is −3.

lim x → 2 x² − 7x + 10x − 2 = −3

5 Check

Let's check with numbers even closer.

x = 1.9999 gives −3.0001.

x = 1.9999: f(x) = −3.0001

x = 2.0001 gives −2.9999.

Still close to −3.

Right!

x = 2.0001: f(x) = −2.9999

Let's look back.

First: try x = 2.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer