Solve f(x) = x^2 - 3 for x < -2, 2x + 5 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 − 3 if x < −2, and 2x + 5 if x ≥ −2.

f(x) = x² − 3 if x < −2

= 2x + 5 if x ≥ −2

1 From the left

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

There, f(x) is x² − 3.

lim x → −2⁻ f(x) = lim x → −2⁻ (x² − 3)

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)² − 3 = 4 − 3 = 1

2 From the right

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

There, f(x) is 2x + 5.

lim x → −2⁺ f(x) = lim x → −2⁺ (2x + 5)

So we use the second piece.

x > −2: piece 2

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

We get 1.

= 2(−2) + 5 = −4 + 5 = 1

3 Compare the sides

From the left we get 1.

From the right we get 1.

left: 1 right: 1

Both sides give the same number.

So the limit is 1.

lim x → −2 f(x) = 1

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 1.

The second piece starts at height 1.

They meet: no jump.

At x = −2 both pieces arrive at 1.

Right!

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Last: check.

Answer