Solve f(x) = 4 - x for x < -1, x^2 + 4 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): 4 − x if x < −1, and x² + 4 if x ≥ −1.

f(x) = 4 − x if x < −1

= x² + 4 if x ≥ −1

1 From the left

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

There, f(x) is 4 − x.

lim x → −1⁻ f(x) = lim x → −1⁻ (4 − x)

So we use the first piece.

x < −1: piece 1

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

We get 5.

= 4 − (−1) = 4 + 1 = 5

2 From the right

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

There, f(x) is x² + 4.

lim x → −1⁺ f(x) = lim x → −1⁺ (x² + 4)

So we use the second piece.

x > −1: piece 2

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

We get 5.

= (−1)² + 4 = 1 + 4 = 5

3 Compare the sides

From the left we get 5.

From the right we get 5.

left: 5 right: 5

Both sides give the same number.

So the limit is 5.

lim x → −1 f(x) = 5

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 5.

The second piece starts at height 5.

They meet: no jump.

At x = −1 both pieces arrive at 5.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer