Is f(x) = (x + 1)/(x - 4) continuous at x = 1 Step by Step

Step by step

The question

Question: decide if f(x) = (x + 1)/(x − 4) is continuous at x = 1.

f(x) = x + 1x − 4

Continuous at x = 1 means three things are true.

One: f(1) has a value.

Two: the limit there exists.

Three: they are equal.

1. f(1) has a value

2. the limit at 1 exists

3. limit = f(1)

1 Does f have a value?

Put 1 in place of x.

f(1) = 1 + 11 − 4 = 2−3

= −23

Careful!

Having a value is not enough.

The graph must also arrive at that height from both sides.

Careful: a value at 1 is not enough

Check 1 is true: f(1) is −2/3.

1. f(1) = −23

2 Does the limit exist?

The bottom is not 0 at 1.

So the limit is just the value: −2/3.

bottom ≠ 0 at 1

So the limit is −2/3.

lim x → 1 f(x) = −23

Check 2 is true: the limit exists.

2. limit = −23

3 Are they equal?

The limit is −2/3, and f(1) is −2/3.

They are equal.

Check 3 is true.

3. limit = −23 = f(1)

All three checks are true.

So f is continuous at x = 1.

= f is continuous at x = 1

4 Check

Let's check on the graph.

The graph is one piece with no gaps.

It passes through (1, −2/3) without lifting the pen.

Right!

Let's look back.

First: does f have a value?.

Then: does the limit exist?.

Then: are they equal?.

Last: check.

Answer