Solve f(x) = x - 1 for x <= 2, x + 1 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): x − 1 if x ≤ 2, and x + 1 if x > 2.

f(x) = x − 1 if x ≤ 2

= x + 1 if x > 2

1 From the left

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

There, f(x) is x − 1.

lim x → 2⁻ f(x) = lim x → 2⁻ (x − 1)

So we use the first piece.

x < 2: piece 1

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

We get 1.

= 2 − 1 = 1

2 From the right

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

There, f(x) is x + 1.

lim x → 2⁺ f(x) = lim x → 2⁺ (x + 1)

So we use the second piece.

x > 2: piece 2

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

We get 3.

= 2 + 1 = 3

3 Compare the sides

From the left we get 1.

From the right we get 3.

left: 1 right: 3

The two sides do not agree.

So the limit does not exist.

lim x → 2 f(x) = does not exist

Careful!

One side is not enough.

1 from the left and 3 from the right do not agree.

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 1.

The second piece starts at height 3.

There is a jump.

At x = 2 the graph jumps from 1 to 3.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer