Solve f(x) = x^2 - 1 for x < 3, 2x + 2 for x >= 3, limit as x -> 3 Step by Step

Step by step

The question

Question: find the limit as x → 3 of this f(x): x² − 1 if x < 3, and 2x + 2 if x ≥ 3.

f(x) = x² − 1 if x < 3

= 2x + 2 if x ≥ 3

1 From the left

From the left means x is a bit less than 3.

There, f(x) is x² − 1.

lim x → 3⁻ f(x) = lim x → 3⁻ (x² − 1)

So we use the first piece.

x < 3: piece 1

A polynomial piece has no gaps, so we just put 3 in.

We get 8.

= 3² − 1 = 9 − 1 = 8

2 From the right

From the right means x is a bit more than 3.

There, f(x) is 2x + 2.

lim x → 3⁺ f(x) = lim x → 3⁺ (2x + 2)

So we use the second piece.

x > 3: piece 2

A polynomial piece has no gaps, so we just put 3 in.

We get 8.

= 2(3) + 2 = 6 + 2 = 8

3 Compare the sides

From the left we get 8.

From the right we get 8.

left: 8 right: 8

Both sides give the same number.

So the limit is 8.

lim x → 3 f(x) = 8

Careful!

One side is not enough.

A limit needs both sides to give the same number.

Careful: both sides must give the same number

4 Check

Let's check on the graph.

The first piece comes in from the left.

It reaches height 8.

The second piece starts at height 8.

They meet: no jump.

At x = 3 both pieces arrive at 8.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer