Is f(x) = x^3 - 2x continuous at x = -1 Step by Step

Step by step

The question

Question: decide if f(x) = x³ − 2x is continuous at x = −1.

f(x) = x³ − 2x

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.

We get 1.

f(−1) = (−1)³ − 2(−1) = −1 + 2 = 1

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 1.

1. f(−1) = 1

2 Does the limit exist?

A polynomial has no gaps and no jumps.

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

polynomial: no gaps, no jumps

So the limit is 1.

lim x → −1 f(x) = 1

Check 2 is true: the limit exists.

2. limit = 1

3 Are they equal?

The limit is 1, and f(−1) is 1.

They are equal.

Check 3 is true.

3. limit = 1 = 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, 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