Is f(x) = x^2 - 3 continuous at x = 2 Step by Step

Step by step

The question

Question: decide if f(x) = x² − 3 is continuous at x = 2.

f(x) = x² − 3

Continuous at x = 2 means three things are true.

One: f(2) has a value.

Two: the limit there exists.

Three: they are equal.

1. f(2) has a value

2. the limit at 2 exists

3. limit = f(2)

1 Does f have a value?

Put 2 in place of x.

We get 1.

f(2) = 2² − 3 = 4 − 3 = 1

Careful!

Having a value is not enough.

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

Careful: a value at 2 is not enough

Check 1 is true: f(2) is 1.

1. f(2) = 1

2 Does the limit exist?

A polynomial has no gaps and no jumps.

So its limit at 2 is just its value: 1.

polynomial: no gaps, no jumps

So the limit is 1.

lim x → 2 f(x) = 1

Check 2 is true: the limit exists.

2. limit = 1

3 Are they equal?

The limit is 1, and f(2) is 1.

They are equal.

Check 3 is true.

3. limit = 1 = f(2)

All three checks are true.

So f is continuous at x = 2.

= f is continuous at x = 2

4 Check

Let's check on the graph.

The graph is one piece with no gaps.

It passes through (2, 1) 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