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

Step by step

The question

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

f(x) = x² if x < 2

= x + 2 if x ≥ 2

1 From the left

From the left means x is a bit less than 2.

There, f(x) is x².

lim x → 2⁻ f(x) = lim x → 2⁻ x²

So we use the first piece.

x < 2: piece 1

A polynomial piece has no gaps, so we just put 2 in.

We get 4.

= 2² = 4

2 From the right

From the right means x is a bit more than 2.

There, f(x) is x + 2.

lim x → 2⁺ f(x) = lim x → 2⁺ (x + 2)

So we use the second piece.

x > 2: piece 2

A polynomial piece has no gaps, so we just put 2 in.

We get 4.

= 2 + 2 = 4

3 Compare the sides

From the left we get 4.

From the right we get 4.

left: 4 right: 4

Both sides give the same number.

So the limit is 4.

lim x → 2 f(x) = 4

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 4.

The second piece starts at height 4.

They meet: no jump.

At x = 2 both pieces arrive at 4.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer