Limit of (x^2 + x - 6)/(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 + x − 6)/(x − 2), using a table.

lim x → 2 x² + x − 6x − 2 =

1 Try x = 2

First, try x = 2 itself.

The top is 0, and the bottom is 0.

2² + 2 − 62 − 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 4.9.

f(1.9) = (1.9)² + (1.9) − 6(1.9) − 2 = 3.61 + 1.9 − 61.9 − 2 = −0.49−0.1 = 4.9

4.9

The others the same way, with a calculator.

4.99

4.999

From the left, f(x) gets close to 5.

from the left: f(x) → 5

3 From the right

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

Work out f(x) for each one.

5.1

5.01

5.001

From the right, f(x) gets close to 5 too.

from the right: f(x) → 5

4 Read the limit

Both sides get close to the same number.

So the limit is 5.

lim x → 2 x² + x − 6x − 2 = 5

5 Check

Let's check with numbers even closer.

x = 1.9999 gives 4.9999.

x = 1.9999: f(x) = 4.9999

x = 2.0001 gives 5.0001.

Still close to 5.

Right!

x = 2.0001: f(x) = 5.0001

Let's look back.

First: try x = 2.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer