Limit of (x^2 + 2x - 3)/(x - 1) as x -> 1 using a table Step by Step

Step by step

The question

Question: find the limit as x → 1 of (x² + 2x − 3)/(x − 1), using a table.

lim x → 1 x² + 2x − 3x − 1 =

1 Try x = 1

First, try x = 1 itself.

The top is 0, and the bottom is 0.

1² + 2(1) − 31 − 1 = 00

We cannot divide by 0.

So f(x) has no value at 1.

00: no value

But a limit does not need the value at 1.

It asks: as x gets close to 1, 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 1: 0.9, 0.99, 0.999.

They come closer and closer to 1.

Put 0.9 in: we get 3.9.

f(0.9) = (0.9)² + 2(0.9) − 3(0.9) − 1 = 0.81 + 1.8 − 30.9 − 1 = −0.39−0.1 = 3.9

3.9

The others the same way, with a calculator.

3.99

3.999

From the left, f(x) gets close to 4.

from the left: f(x) → 4

3 From the right

Now numbers just above 1: 1.1, 1.01, 1.001.

Work out f(x) for each one.

4.1

4.01

4.001

From the right, f(x) gets close to 4 too.

from the right: f(x) → 4

4 Read the limit

Both sides get close to the same number.

So the limit is 4.

lim x → 1 x² + 2x − 3x − 1 = 4

5 Check

Let's check with numbers even closer.

x = 0.9999 gives 3.9999.

x = 0.9999: f(x) = 3.9999

x = 1.0001 gives 4.0001.

Still close to 4.

Right!

x = 1.0001: f(x) = 4.0001

Let's look back.

First: try x = 1.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer