Limit of (x^2 - 3x - 4)/(x - 4) as x -> 4 using a table Step by Step

Step by step

The question

Question: find the limit as x → 4 of (x² − 3x − 4)/(x − 4), using a table.

lim x → 4 x² − 3x − 4x − 4 =

1 Try x = 4

First, try x = 4 itself.

The top is 0, and the bottom is 0.

4² − 3(4) − 44 − 4 = 00

We cannot divide by 0.

So f(x) has no value at 4.

00: no value

But a limit does not need the value at 4.

It asks: as x gets close to 4, 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 4: 3.9, 3.99, 3.999.

They come closer and closer to 4.

Put 3.9 in: we get 4.9.

f(3.9) = (3.9)² − 3(3.9) − 4(3.9) − 4 = 15.21 − 11.7 − 43.9 − 4 = −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 4: 4.1, 4.01, 4.001.

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 → 4 x² − 3x − 4x − 4 = 5

5 Check

Let's check with numbers even closer.

x = 3.9999 gives 4.9999.

x = 3.9999: f(x) = 4.9999

x = 4.0001 gives 5.0001.

Still close to 5.

Right!

x = 4.0001: f(x) = 5.0001

Let's look back.

First: try x = 4.

Then: from the left.

Then: from the right.

Then: read the limit.

Last: check.

Answer