Solve f(x) = x^2 for x <= 1, x + 3 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 if x ≤ 1, and x + 3 if x > 1.

f(x) = x² if x ≤ 1

= x + 3 if x > 1

1 From the left

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

There, f(x) is x².

lim x → 1⁻ f(x) = lim x → 1⁻ x²

So we use the first piece.

x < 1: piece 1

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

We get 1.

= 1² = 1

2 From the right

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

There, f(x) is x + 3.

lim x → 1⁺ f(x) = lim x → 1⁺ (x + 3)

So we use the second piece.

x > 1: piece 2

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

We get 4.

= 1 + 3 = 4

3 Compare the sides

From the left we get 1.

From the right we get 4.

left: 1 right: 4

The two sides do not agree.

So the limit does not exist.

lim x → 1 f(x) = does not exist

Careful!

One side is not enough.

1 from the left and 4 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 4.

There is a jump.

At x = 1 the graph jumps from 1 to 4.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer