Solve f(x) = x^2 + 2x for x < 1, 5 - 2x for x >= 1, limit as x -> 1 Step by Step

Step by step

The question

Question: find the limit as x → 1 of this f(x): x² + 2x if x < 1, and 5 − 2x if x ≥ 1.

f(x) = x² + 2x if x < 1

= 5 − 2x if x ≥ 1

1 From the left

From the left means x is a bit less than 1.

There, f(x) is x² + 2x.

lim x → 1⁻ f(x) = lim x → 1⁻ (x² + 2x)

So we use the first piece.

x < 1: piece 1

A polynomial piece has no gaps, so we just put 1 in.

We get 3.

= 1² + 2(1) = 1 + 2 = 3

2 From the right

From the right means x is a bit more than 1.

There, f(x) is 5 − 2x.

lim x → 1⁺ f(x) = lim x → 1⁺ (5 − 2x)

So we use the second piece.

x > 1: piece 2

A polynomial piece has no gaps, so we just put 1 in.

We get 3.

= 5 − 2(1) = 5 − 2 = 3

3 Compare the sides

From the left we get 3.

From the right we get 3.

left: 3 right: 3

Both sides give the same number.

So the limit is 3.

lim x → 1 f(x) = 3

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 3.

The second piece starts at height 3.

They meet: no jump.

At x = 1 both pieces arrive at 3.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer